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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video .
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did n't sal say in previous videos that theta should be between - pi/2 and pi/2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle .
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when he says that it looks more like an ellipse what is an ellipse , please ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere .
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: how is arccosine ( -1 ) =0 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta .
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why is the arcsin or arccos of whole numbers is math error ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there .
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why is the arccos x=o , the o can only greater than 0 and less than 180 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ?
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why is n't less than 360 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 .
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would someone mind explaining how those values were deduced ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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sal explains that cos ( arccos x ) = x and sin ( arcsin x ) = x , but what about tan ( arctan x ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle .
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can someone explain how sal gets to 3/4 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle .
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what is the main difference between arccos , arcsin , arctan and sec , csc , cot ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta .
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so are the arcsin , arccos , and arctan like functions to find angles corresponding to certain given values ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ?
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is n't it suppose to equal 150 degrees since 30 degrees is opposite the side -1/2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there .
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should n't it be 180 degrees - 30 degrees since the 1/2 is opposite the 30 degree angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ?
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why does sal write the restiction of arccos in radians ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta .
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can we restrict the range to the lower half-circle , so ( pi < = theta < = 2pi ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 .
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while the range of cos is between -1 and 1 therefore the domain of arccos can only be between -1 and 1 and therefore the range of arccos can only be between 0 and pi radians ( or 0 and 180 degrees ) where 's the whole restriction from ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi .
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why the radiant range is equal ( 0 : pi ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video .
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when sal wants to restrict the domain , should n't the domain be 0 < theta < pi/2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea .
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pi translates to a full circle , but he wants to restrict the domain to half a circle , correct ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x .
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why is the domain of the theta of arcccos ( costheta ) = ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 .
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what is arcos ( cos ( -30 ) ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there .
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is there a difference between sin ( 0 ) = ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x .
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how do you mathematically find theta ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times .
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what if theta is negative ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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why do you have to restrict thr domain and the range ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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why does the range have to be specific to the upper quadrants and not the whole circle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 .
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how does sal know what the function is restricted to ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees .
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can someone please explain the difference between inverse and reciprocal ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ?
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can the value of cos ( arccos x ) be equal to the x value if x is not in the domain ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do .
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how do i understand which quadrant does -1/2 belong to and how do i understand whether to take it on the x axis or the y axis ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees .
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why do you need to find out the angle supplementary to 60 degrees ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi .
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when he drew the unit circle to find arc cos ( -1/2 ) why did use the `` big '' angle over the little one ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants .
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on what basis do we define the range of sin cos or tan ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ?
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why is theta the 120 degree angle instead of the 60 degree angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here .
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would n't cos ( 60 ) be equal to 1/2 or -1/2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out .
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how do yo know whether the angle they want you to figure out is 60 or 120 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi .
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at minute sal says that cos of theta = -1/2 , is n't it cos^-1 of theta = -1/2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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so why does arccosine use the first and second quadrants , whereas arctangent and arcsine use the first and fourth ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 .
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how do i determine arcsin/arccos of a triangle that is not 30-60-90 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians .
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lets supose i dont know that the angle corresponding to sqrt ( 3 ) /2 is 60 degrees , how can i search 2pi/3 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine .
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is arccos always restricted to quadrants one and two ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi .
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in the unit circle that sal drew , why is n't -1/2 the cosine of 60 degrees ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle .
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arcsin vs cosecant what is the difference ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 .
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are the range for arccosine always quadrants i and ii ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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how would you solve sin ( 3arccos ( x ) ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ?
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is it possible to restrict range before obtaining the angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question .
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why is arccos not able to be in q3 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta .
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why do we limit the range of our values for theta ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer .
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so but how do we answer the question `` what is the principal value of arctan = -44 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ?
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why is the arccos ( -1/2 ) 120 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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can your range include the other hemisphere/domain instead ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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can your range include the other hemisphere/domain instead ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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can your range include the other hemisphere/domain instead ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video .
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is arccos +ve in the [ -pi/2 , 0 ) interval ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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why is the domain restricted to the 1st and 2nd quadrants but it was restricted to the 1st and 4th for sin ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process .
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how do you choose the restrictions ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this .
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how to make arcsin ( sinx ) graph ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video .
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how would you solve for y = arccos ( x^2 ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta .
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how do you limit the domain of inverse trig functions ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere .
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what 's the difference between arccosine and secant ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer .
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i do n't understand , why is sal restricting the values , why does n't he want the second value within a single rotation ( 2pi ) ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top .
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why do we restrict the range to quadrants i and ii ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x .
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what is the principal value of a trig function ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta .
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why is the range always 0 < = cos or arccos of theta < = pi ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea .
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what if we have some kind of value such as arcsin ( 2/5 ) with which we ca n't use the unit circle as a reference ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea .
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this point is also in the 3rd quadrant ( 4pi/3 ) , so how is it that we know that the angle points `` up '' on the unit circle and not `` down '' ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea .
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why use a unit circle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians .
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why not use a circle with radius 2 or 3 or 5 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians .
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do the trignometric ratios hold good for any circle of radius maybe 2,3,4,5 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea .
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how do u determine which angle u have to find on the unit circle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger .
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why do we have to find the light blue angle instead of the green angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video .
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i do n't get it ... cos is a/h so if the cos results in -1/2 does n't that mean the adjacent side is -1 and the hypotenuse and therefore radius is 2 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 .
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with the first triangle , how did sal get the 60 degree angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here .
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how do you find the different ranges for arctan and arccos ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ?
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why do you find the outside supplementary angle for arccos but the inside negative angle on arcsin ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta .
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are the trigonometric identities of arcsine arccosine and arctangent ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ?
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is there a way to calculate this angle without first using degrees ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on .
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why we restrict ourselves to 1st nd 2nd quadrant ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one .
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i 'm still confused ... what would the arccos1/sqrt3 be ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ?
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when he does the first problem , why is n't theta the 60 degree angle ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there .
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why is arccos ( 0 ) 90 degrees if a triangle cant be made when x = 0 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 .
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why is the cos of 3pi -1 ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle .
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what is the difference between arccos and sec ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 .
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why do we call the inverse of sin , cos , and tan : arcsin and arccos and arctan ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x .
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why sine is y and cosine is x ?
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i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so i 've already made videos on the arcsine and the arctangent , so to kind of complete the trifecta , i might as well make a video on the arccosine . and just like the other inverse trigonometric functions , the arccosine is kind of the same thought process . if i were to tell you the arc , no , i 'm doing cosine , if our tell you that arccosine of x is equal to theta . this is an equivalent statement to saying that the inverse cosine of x is equal to theta . these are just two different ways of writing the exact same thing . and as soon as i see either an arc- anything , or an inverse trig function in general , my brain immediately rearranges this . my brain immediately says , this is saying that if i take the cosine of some angle theta , that i 'm going to get x . or that same statement up here . either of these should boil down to this . if i say , you know , what is the inverse cosine of x , my brain says , what angle can i take the cosine of to get x ? so with that said , let 's try it out on an example . let 's say that i have the arc , i 'm told , no , two c 's there , i 'm told to evaluate the arccosine of minus 1/2 . my brain , you know , let 's say that this is going to be equal to , it 's going to be equal to some angle . and this is equivalent to saying that the cosine of my mystery angle is equal to minus 1/2 . and as soon as you put it in this way , at least for my brain , it becomes a lot easier to process . so let 's draw our unit circle and see if we can make some headway here . so that 's my , let me see if i can draw a little straighter . maybe i could actually draw , put rulers here , and if i put a ruler here , maybe i can draw a straight line . let me see . no , that 's too hard . ok , so that is my y-axis , that is my x-axis . not the most neatly drawn axes ever , but it 'll do . let me draw my unit circle . looks more like a unit ellipse , but you get the idea . and the cosine of an angle as defined on the unit circle definition is the x-value on the unit circle . so if we have some angle , the x-value is going to be equal a minus 1/2 . so we got a minus 1/2 right here . and so the angle that we have to solve for , our theta , is the angle that when we intersect the unit circle , the x-value is minus 1/2 . so let me see , this is the angle that we 're trying to figure out . this is theta that we need to determine . so how can we do that ? so this is minus 1/2 right here . let 's figure out these different angles . and the way i like to think about it is , i like to figure out this angle right here . and if i know that angle , i can just subtract that from 180 degrees to get this light blue angle that 's kind of the solution to our problem . so let me make this triangle a little bit bigger . so that triangle , let me do it like this . that triangle looks something like this . where this distance right here is 1/2 . that distance right there is 1/2 . this distance right here is 1 . hopefully you recognize that this is going to be a 30 , 60 , 90 triangle . you could actually solve for this other side . you 'll get the square root of 3 over 2 . and to solve for that other side you just need to do the pythagorean theorem . actually , let me just do that . let me just call this , i do n't know , just call this a . so you 'd get a squared , plus 1/2 squared , which is 1/4 , which is equal to 1 squared , which is 1 . you get a squared is equal to 3/4 , or a is equal to the square root of 3 over 2 . so you immediately know this is a 30 , 60 , 90 triangle . and you know that because the sides of a 30 , 60 , 90 triangle , if the hypotenuse is 1 , are 1/2 and square root of 3 over 2 . and you also know that the side opposite the square root of 3 over 2 side is 60 degrees . that 's 60 , this is 90 . this is the right angle , and this is 30 right up there . but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . so the arccosine , or the inverse cosine , let me write that down . the arccosine of minus 1/2 is equal to 120 degrees . did i write 180 there ? no , it 's 180 minus the 60 , this whole thing is 180 , so this is , right here is , 120 degrees , right ? 120 plus 60 is 180 . or , if we wanted to write that in radians , you just right 120 degrees times pi radian per 180 degrees , degrees cancel out . 12 over 18 is 2/3 , so it equals 2 pi over 3 radians . so this right here is equal to 2 pi over 3 radians . now , just like we saw in the arcsine and the arctangent videos , you probably say , hey , ok , if i have 2 pi over 3 radians , that gives me a cosine of minus 1/2 . and i can write that . cosine of 2 pi over 3 is equal to minus 1/2 . this gives you the same information as this statement up here . but i can just keep going around the unit circle . for example , i could , how about this point over here ? cosine of this angle , if i were to add , if i were to go this far , would also be minus 1/2 . and then i could go 2 pi around and get back here . so there 's a lot of values that if i take the cosine of those angles , i 'll get this minus 1/2 . so we have to restrict ourselves . we have to restrict the values that the arccosine function can take on . so we 're essentially restricting it 's range . we 're restricting it 's range . what we do is we restrict it 's range to this upper hemisphere , the first and second quadrants . so if we say , if we make the statement that the arccosine of x is equal to theta , we 're going to restrict our range , theta , to that top . so theta is going to be greater than or equal to 0 and less than or equal to 2 pi . less , oh sorry , not 2 pi . less than or equal to pi , right ? where this is also 0 degrees , or 180 degrees . we 're restricting ourselves to this part of the hemisphere right there . and so you ca n't do this , this is the only point where the cosine of the angle is equal minus 1/2 . we ca n't take this angle because it 's outside of our range . and what are the valid values for x ? well any angle , if i take the cosine of it , it can be between minus 1 and plus 1 . so x , the domain for the arccosine function , is going to be x has to be less than or equal to 1 and greater than or equal to minus 1 . and once again , let 's just go check our work . let 's see if the value i got here , that the arccosine of minus 1/2 really is 2 pi over 3 as calculated by the ti-85 . we turn it on . so i need to figure out the inverse cosine , which is the same thing as the arccosine of minus 1/2 , of minus 0.5 . it gives me that decimal , that strange number . let 's see if that 's the same thing as 2 pi over 3 . 2 times pi divided by 3 is equal to , that exact same number . so the calculator gave me the same value i got . but this is kind of a useless , well , it 's not a useless number . it 's a valid , that is the answer . but it does n't , it 's not a nice clean answer . i did n't know that this is 2 pi over 3 radians . and so when we did it using the unit circle , we were able to get that answer . so hopefully , actually let me ask you , let me just finish this up with an interesting question . and this applies to all of them . if i were to ask you , you know , say i were to take the arccosine of x , and then i were to take the cosine of that , what is this going to be equal to ? well , this statement right here can be said , well , let 's say that the arccosine of x is equal to theta , that means that the cosine of theta is equal to x , right ? so if the arccosine of x is equal to theta , we can replace this with theta . and then the cosine of theta , well the cosine of theta is x . so this whole thing is going to be x. hopefully i did n't get confuse you there , right ? i 'm saying look , arccosine of x , just call that theta . now , by definition , this means that the cosine of theta is equal to x . these are equivalent statements . these are completely equivalent statements right here . so if we put a theta right there , we take the cosine of theta , it has to be equal to x . now let me ask you a bonus , slightly trickier question . what if i were to ask you , and this is true for any x that you put in here . this is true for any x , any value between negative 1 and 1 including those two endpoints , this is going to be true . now what if i were ask you what the arccosine of the cosine of theta is ? what is this going to be equal to ? my answer is , it depends on the theta . so , if theta is in the , if theta is in the range , if theta is between , if theta is between 0 and pi , so it 's in our valid a range for , kind of , our range for the product of the arccosine , then this will be equal to theta . if this is true for theta . but what if we take some theta out of that range ? let 's try it out . let 's take , so let me do one with theta in that range . let 's take the arccosine of the cosine of , let 's just do one of them that we know . let 's take the cosine of , let 's stick with cosine of 2 pi over 3 . cosine of 2 pi over 3 radians , that 's the same thing as the arccosine of minus 1/2 . cosine of 2 pi over 3 is minus 1/2 . we just saw that in the earlier part of this video . and then we solved this . we said , oh , this is equal to 1 pi over 3 . so for in the range of thetas between 0 and pi it worked . and that 's because the arccosine function can only produce values between 0 and pi . but what if i were to ask you , what is the arccosine of the cosine of , i do n't know , of 3 pi . so if i were to draw the unit circle here , let me draw the unit circle , a real quick one . and that 's my axes . what 's 3 pi ? 2 pi is if i go around once . and then i go around another pi , so i end up right here . so i 've gone around 1 1/2 times the unit circle . so this is 3 pi . what 's the x-coordinate here ? it 's minus 1 . so cosine of 3 pi is minus 1 , right ? so what 's arccosine of minus 1 ? arccosine of minus 1 . well remember , the range , or the set of values , that arccosine can evaluate to is in this upper hemisphere . it 's between , this can only be between pi and 0 . so arccosine of negative 1 is just going to be pi . so this is going to be pi . arccosine of negative , this is negative 1 , arccosine of negative 1 is pi . and that 's a reasonable statement , because the difference between 3 pi and pi is just going around the unit circle a couple of times . and so you get an equivalent , it 's kind of , you 're at the equivalent point on the unit circle . so i just thought i would throw those two at you . this one , i mean this is a useful one . well , actually , let me write it up here . this one is a useful one . the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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the cosine of the arccosine of x is always going to be x. i could also do that with sine . the sine of the arcsine of x is also going to be x . and these are just useful things to , you should n't just memorize them , because obviously you might memorize it the wrong way , but you should just think a little bit about it , and you 'll never forget it .
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so is it accurate to state that cos cancels out arccos in the statement `` cos ( arccos x ) = x '' ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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i dont understand , like why does the y axis have to have the price , and the x axis is months why cant the y axis be months , and x axis be price ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
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what is the difference between a line graph and a broken line graph ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question .
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why are more lines sometimes used if the number is increasing or decreasing ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month .
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would n't it make more sense to use continuous data like temperature or elevation instead of discrete data like the number of sweatshirts ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ?
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does everything in the world use math ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
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can a line graph possibly be more helpful than bar graph ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
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when you are doing math can you use a graph to graph slopes on a coordinate grid ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month .
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how do we know that the points is 11 ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
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how does a line graph what ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ?
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why does everything use so much math in the world ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ?
|
where do learn about mapping of orderer pairs and how to develop sequences of numbers from diagrams and contexts ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 .
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does the subject go on the x axis or y axis ?
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an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
|
and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
|
when you are doing math can you use a graph to graph slopes on a coordinate grid ?
|
an investment firm creates a graph showing the performance of a specific stock over 12 months . over the course of the year , is the price of the stock rising , falling , or staying the same ? so over this axis right over here , the horizontal axis , they have month by month . and we move forward in time -- july , august , september , october . and in this axis , the vertical axis , we have the price . so , for example , in july the price of this stock was a little over $ 10 . then in august , it moved up to -- it looks like around $ 11 . and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line . and the reason why we connect them with a line is to really see if there 's some kind of a trend here to really show that you have something that 's moving from one price to another . and so line graphs tend to be used when you have something that 's changing over time . now , with that out of the way , let 's actually answer their question . over the course of the year , is the price of the stock rising , falling , or staying the same ? so on a month-to-month basis , you have , for example , from july to august , the price went up . then from august to september , the price went down . then it went up for two months . then it went down for a month . then it went up for a couple more months . then it went really up from february to march , went all the way up to almost $ 17 . then it went down again . and then it kept going up again . but they 're asking us not did it go up every month . they 're saying over the course of the year is the price of the stock rising , falling , or staying the same ? and if you go from july , which is where our data starts right over here , our price was around $ 10 . and even though there were a few months where it went down , the overall trend is that the price is going up . the overall trend is that the price going up . and you can even see that . in july , it was $ 10 . and then by june of the next year , it was approaching -- i do n't know , it looks like it 's about a little over $ 16 , maybe almost $ 17 . so it actually had gone up a lot . they do n't give us july of the next year . but the overall trend is definitely the upwards direction right over here . and you can see it just visually by looking at this line graph . even though there 's a few bumps that go down , the overall direction is upward .
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and then we could keep going month by month . and this type of graph right over here is called a line graph because you have the data points for each month . and then we connected them with a line .
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how do you calculate that graph ?
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