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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 trigonomet... | 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 . | did n't sal say in previous videos that theta should be between - pi/2 and pi/2 ? |
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 trigonomet... | 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 . | when he says that it looks more like an ellipse what is an ellipse , please ? |
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 trigonomet... | 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 . | : how is arccosine ( -1 ) =0 ? |
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 trigonomet... | 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 . | why is the arcsin or arccos of whole numbers is math error ? |
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 trigonomet... | 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 . | why is the arccos x=o , the o can only greater than 0 and less than 180 ? |
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 trigonomet... | 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 ? | why is n't less than 360 ? |
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 trigonomet... | 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 . | would someone mind explaining how those values were deduced ? |
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 trigonomet... | 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 i... | sal explains that cos ( arccos x ) = x and sin ( arcsin x ) = x , but what about tan ( arctan x ) ? |
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 trigonomet... | 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 . | can someone explain how sal gets to 3/4 ? |
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 trigonomet... | 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 . | what is the main difference between arccos , arcsin , arctan and sec , csc , cot ? |
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 trigonomet... | 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 trigonomet... | so are the arcsin , arccos , and arctan like functions to find angles corresponding to certain given values ? |
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 trigonomet... | 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 ? | is n't it suppose to equal 150 degrees since 30 degrees is opposite the side -1/2 ? |
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 trigonomet... | 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 . | should n't it be 180 degrees - 30 degrees since the 1/2 is opposite the 30 degree angle ? |
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 trigonomet... | 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 ? | why does sal write the restiction of arccos in radians ? |
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 trigonomet... | 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 . | can we restrict the range to the lower half-circle , so ( pi < = theta < = 2pi ) ? |
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 trigonomet... | 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 . | 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 ? |
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 trigonomet... | 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 . | why the radiant range is equal ( 0 : pi ) ? |
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 trigonomet... | 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 . | when sal wants to restrict the domain , should n't the domain be 0 < theta < pi/2 ? |
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 trigonomet... | 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 . | pi translates to a full circle , but he wants to restrict the domain to half a circle , correct ? |
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 trigonomet... | 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 . | why is the domain of the theta of arcccos ( costheta ) = ? |
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 trigonomet... | 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 . | what is arcos ( cos ( -30 ) ) ? |
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 trigonomet... | 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 . | is there a difference between sin ( 0 ) = ? |
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 trigonomet... | 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 . | how do you mathematically find theta ? |
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 trigonomet... | 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 . | what if theta is negative ? |
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 trigonomet... | 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 ... | why do you have to restrict thr domain and the range ? |
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 trigonomet... | 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 . | why does the range have to be specific to the upper quadrants and not the whole circle ? |
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 trigonomet... | 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 . | how does sal know what the function is restricted to ? |
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 trigonomet... | 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 . | can someone please explain the difference between inverse and reciprocal ? |
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 trigonomet... | 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 th... | can the value of cos ( arccos x ) be equal to the x value if x is not in the domain ? |
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 trigonomet... | 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 . | 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 ? |
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 trigonomet... | what 's the bigger angle that we care about ? what is 60 degrees supplementary to ? it 's supplementary to 180 degrees . | why do you need to find out the angle supplementary to 60 degrees ? |
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 trigonomet... | 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 . | when he drew the unit circle to find arc cos ( -1/2 ) why did use the `` big '' angle over the little one ? |
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 trigonomet... | 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 . | on what basis do we define the range of sin cos or tan ? |
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 trigonomet... | but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? | why is theta the 120 degree angle instead of the 60 degree angle ? |
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 trigonomet... | 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 . | would n't cos ( 60 ) be equal to 1/2 or -1/2 ? |
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 trigonomet... | 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 . | how do yo know whether the angle they want you to figure out is 60 or 120 ? |
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 trigonomet... | 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 . | at minute sal says that cos of theta = -1/2 , is n't it cos^-1 of theta = -1/2 ? |
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 trigonomet... | 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 why does arccosine use the first and second quadrants , whereas arctangent and arcsine use the first and fourth ? |
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 trigonomet... | 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 . | how do i determine arcsin/arccos of a triangle that is not 30-60-90 ? |
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 trigonomet... | 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 . | lets supose i dont know that the angle corresponding to sqrt ( 3 ) /2 is 60 degrees , how can i search 2pi/3 ? |
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 trigonomet... | 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 . | is arccos always restricted to quadrants one and two ? |
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 trigonomet... | 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 . | in the unit circle that sal drew , why is n't -1/2 the cosine of 60 degrees ? |
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 trigonomet... | 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 . | arcsin vs cosecant what is the difference ? |
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 trigonomet... | 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 . | are the range for arccosine always quadrants i and ii ? |
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 trigonomet... | 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 i... | how would you solve sin ( 3arccos ( x ) ) ? |
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 trigonomet... | 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 ? | is it possible to restrict range before obtaining the angle ? |
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 trigonomet... | 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 . | why is arccos not able to be in q3 ? |
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 trigonomet... | 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 . | why do we limit the range of our values for theta ? |
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 trigonomet... | 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 . | so but how do we answer the question `` what is the principal value of arctan = -44 ? |
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 trigonomet... | 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 ? | why is the arccos ( -1/2 ) 120 ? |
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 trigonomet... | 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 ... | can your range include the other hemisphere/domain instead ? |
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 trigonomet... | 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 ... | can your range include the other hemisphere/domain instead ? |
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 trigonomet... | 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 ... | can your range include the other hemisphere/domain instead ? |
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 trigonomet... | 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 . | is arccos +ve in the [ -pi/2 , 0 ) interval ? |
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 trigonomet... | 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 . | why is the domain restricted to the 1st and 2nd quadrants but it was restricted to the 1st and 4th for sin ? |
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 trigonomet... | 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 trigonomet... | how do you choose the restrictions ? |
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 trigonomet... | 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 . | how to make arcsin ( sinx ) graph ? |
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 trigonomet... | 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 . | how would you solve for y = arccos ( x^2 ) ? |
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 trigonomet... | 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 trigonomet... | how do you limit the domain of inverse trig functions ? |
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 trigonomet... | 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 . | what 's the difference between arccosine and secant ? |
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 trigonomet... | 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 . | 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 ) ? |
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 trigonomet... | 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 . | why do we restrict the range to quadrants i and ii ? |
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 trigonomet... | 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 . | what is the principal value of a trig function ? |
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 trigonomet... | 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 . | why is the range always 0 < = cos or arccos of theta < = pi ? |
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 trigonomet... | 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 . | 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 ? |
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 trigonomet... | 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 . | 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 '' ? |
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 trigonomet... | 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 . | why use a unit circle ? |
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 trigonomet... | 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 . | why not use a circle with radius 2 or 3 or 5 ? |
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 trigonomet... | 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 . | do the trignometric ratios hold good for any circle of radius maybe 2,3,4,5 ? |
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 trigonomet... | 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 . | how do u determine which angle u have to find on the unit circle ? |
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 trigonomet... | 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 . | why do we have to find the light blue angle instead of the green angle ? |
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 trigonomet... | 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 . | 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 ? |
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 trigonomet... | 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 . | with the first triangle , how did sal get the 60 degree angle ? |
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 trigonomet... | 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 . | how do you find the different ranges for arctan and arccos ? |
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 trigonomet... | 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 ? | why do you find the outside supplementary angle for arccos but the inside negative angle on arcsin ? |
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 trigonomet... | 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 trigonomet... | are the trigonometric identities of arcsine arccosine and arctangent ? |
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 trigonomet... | but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? | is there a way to calculate this angle without first using degrees ? |
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 trigonomet... | 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 . | why we restrict ourselves to 1st nd 2nd quadrant ? |
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 trigonomet... | 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 . | i 'm still confused ... what would the arccos1/sqrt3 be ? |
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 trigonomet... | but this is the one we care about . this angle right here we just figured out is 60 degrees . so what 's this ? | when he does the first problem , why is n't theta the 60 degree angle ? |
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 trigonomet... | 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 . | why is arccos ( 0 ) 90 degrees if a triangle cant be made when x = 0 ? |
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 trigonomet... | 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 . | why is the cos of 3pi -1 ? |
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 trigonomet... | 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 . | what is the difference between arccos and sec ? |
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 trigonomet... | 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 . | why do we call the inverse of sin , cos , and tan : arcsin and arccos and arctan ? |
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 trigonomet... | 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 . | why sine is y and cosine is x ? |
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 trigonomet... | 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 i... | so is it accurate to state that cos cancels out arccos in the statement `` cos ( arccos x ) = x '' ? |
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 , augu... | 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 . | 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 ? |
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 , augu... | 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 . | what is the difference between a line graph and a broken line graph ? |
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 , augu... | 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... | why are more lines sometimes used if the number is increasing or decreasing ? |
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 , augu... | 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 . | would n't it make more sense to use continuous data like temperature or elevation instead of discrete data like the number of sweatshirts ? |
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 , augu... | 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 ? | does everything in the world use math ? |
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 , augu... | 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 . | can a line graph possibly be more helpful than bar graph ? |
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 , augu... | 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 , augu... | 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 . | how do we know that the points is 11 ? |
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 , augu... | 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 . | how does a line graph what ? |
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 , augu... | 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 ? | why does everything use so much math in the world ? |
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 , augu... | 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 ? |
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 , augu... | 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 . | does the subject go on the x axis or y axis ? |
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 , augu... | 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 , augu... | 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 . | how do you calculate that graph ? |
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