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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great .
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why , then , do some of the vectors point towards contour lines that are closer together ( which makes sense because close contour lines indicate high steepness ) , yet in another section of the field the vectors are pointing toward contour lines that are farther apart ?
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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this .
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how to plot vector fields in microsoft mathematics ?
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so here i want to talk about the gradient and the context of a contour map . so let 's say we have a multivariable function . a two-variable function f of x , y . and this one is just gon na equal x times y . so we can visualize this with a contour map just on the xy plane . so what i 'm gon na do is i 'm gon na go over here . i 'm gon na draw a y axis and my x axis . all right , so this right here represents x values . and this represents y values . and this is entirely the input space . and i have a video on contour maps if you are unfamiliar with them or are feeling uncomfortable . and the contour map for x times y looks something like this . and each one of these lines represents a constant value . so you might be thinking that you have , you know , let 's say you want a the constant value for f of x times y is equal to two . would be one of these lines . that would be what one of these lines represents . and a way you could think about that for this specific function is you saying hey , when is x times y equal to two ? and that 's kind of like the graph y equals two over x . and that 's where you would see something like this . so all of these lines , they 're representing constant values for the function . and now i want to take a look at the gradient field . and the gradient , if you 'll remember , is just a vector full of the partial derivatives of f. and let 's just actually write it out . the gradient of f , with our little del symbol , is a function of x and y . and it 's a vector-valued function whose first coordinate is the partial derivative of f with respect to x . and the second component is the partial derivative with respect to y . so when we actually do this for our function , we take the partial derivative with respect to x . it takes a look . x looks like a variable . y looks like a constant . the derivative of this whole thing is just equal to that constant , y . and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well . you know , at every given point , xy , so you kind of go like x equals two , y equals one , let 's say . so that would be x equals two , y equals one . you would plug in the vector and see what should be output . and at this point , the point is two , one . the desired output kind of swaps those . so we 're looking somehow to draw the vector one , two . so you would expect to see the vector that has an x component of one and a y component of two . something like that . but it 's probably gon na be scaled down because of the way we usually draw vector fields . and the entire field looks like this . so i 'll go ahead and erase what i had going on . since this is a little bit clearer . and remember , we scaled down all the vectors . the color represents length . so red here is super-long . blue is gon na be kind of short . and one thing worth noticing . if you take a look at all of the given points around , if the vector is crossing a contour line , it 's perpendicular to that contour line . wherever you go . you know , you go down here , this vector 's perpendicular to the contour line . over here , perpendicular to the contour line . and this happens everywhere . and it 's for a very good reason . and it 's also super-useful . so let 's just think about what that reason should be . let 's zoom in on a point . so i 'm gon na clear up our function here . clear up all of the information about it . and just zoom in on one of those points . so let 's say like right here . we 'll take that guy and kind of imagine zooming in and saying what 's going on in that region ? so you 've got some kind of contour line . and it 's swooping down like this . and that represents some kind of value . let 's say that represents the value f equals two . and , you know , it might not be a perfect straight line . but the more you zoom in , the more it looks like a straight line . and when you want to interpret the gradient vector . if you remember , in the video about how to interpret the gradient in the context of a graph , i said it points in the direction of steepest descent . so if you imagine all the possible vectors kind of pointing away from this point , the question is , which direction should you move to increase the value of f the fastest ? and there 's two ways of thinking about that . one is to look at all of these different directions and say which one increases x the most ? but another way of doing it would be to get rid of them all and just take a look at another contour line that represents a slight increase . all right , so let 's say you 're taking a look at a contour line , another contour line . something like this . and maybe that represents something that 's right next to it . like 2.1 . that represents , you know , another value that 's very close . and if i were a better artist , and this was more representative , it would look like a line that 's parallel to the original one . because if you change the output by just a little bit , the set of in points that look like it is pretty much the same but just shifted over a bit . so another way we can think about the gradient here is to say of all of the vectors that move from this output of two up to the value of 2.1 . you know , you 're looking at all of the possible different vectors that do that . you know , which one does it the fastest ? and this time , instead of thinking of the fastest as constant-length vectors , what increases it the most , we 'll be thinking , constant increase in the output . which one does it with the shortest distance ? and if you think of them as being roughly parallel lines , it should n't be hard to convince yourself that the shortest distance is n't gon na be , you know , any of those . it 's gon na be the one that connects them pretty much perpendicular to the original line . because if you think about these as lines , and the more you zoom in , the more they pretty much look like parallel lines , the path that connects one to the other is gon na be perpendicular to both of them . so because of this interpretation of the gradient as the direction of steepest descent , it 's a natural consequence that every time it 's on a contour line , wherever you 're looking it 's actually perpendicular to that line . because you can think about it as getting to the next contour line as fast as it can . increasing the function as fast as it can . and this is actually a very useful intepretation of the gradient in different contexts . so it 's a good one to keep in the back of your mind . gradient is always perpendicular to contour lines . great . see you next video .
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and then kind of the reverse for when you take the partial derivative with respect to y. y looks like a variable . x looks like a constant . and the derivative is just that constant , x . and this can be visualized as a vector field in the xy plane as well .
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using the power rule i thought the derivative of a constant was 0 why does he say it 's one ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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do you have to take the psat test before the sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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does the new sat include pre-calculus math problems ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life .
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would taking pre-calculus in junior year benefit a student taking the sat at the end of 11th grade ( vs taking algebra 2 during junior year ) ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what age do you have to be to take the sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what 's the difference between sat i & sat ii ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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will i be needing to take an sat anytime soon ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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for the new sat , should i spend time studying a load of sat vocab like i did for the last sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college .
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what do i do if i took algebra 1 in grade 9 , geometry in grade 10 , algebra 2 in 11th grade , and going to take pre calc in 12th grade ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this .
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how will i be fully prepared for the sat if i do n't know some of the math that will be on the test ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast .
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should i take the writing section of the sat and , if so , why is it better to do the optional writing part ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that .
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is there going be formula 's on the new sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast .
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the writing part , is there any way to take an sat and still take the writing part ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat .
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is the history section mandatory for people who want to take science ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what 's the difference between the act and the sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
|
so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what types of math should i do to take the sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
|
so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what is the sat equivalent in the uk ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
|
so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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can i take two sat tests ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this .
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i know it is suggested that one should not sit down and just memorize numerous vocabulary words for the new sat , but what should i do about this issue of mine ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that .
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is the optional essay one that is highly recommended for college applications ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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is there the part of `` sentence completion '' in the new sat ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra .
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why is the sat so important to get into a college ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well .
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is it monitory to do the writing portion on the test as a avid student ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it .
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will we always have access to all the questions and practices ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life .
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do you have to take both the sat and the act to get into a college ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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should a student that is a junior in high school ( who takes algebra 2 full year ) taking the new sat this year be worried about the calculus questions on the math sections ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ?
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when i take the sat test , can i write/underline/make marks on it ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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what does sat stand for ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty .
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in the psat does the new rule about `` no penalties for wrong answers '' still apply ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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why did the sat change ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
|
so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that .
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do you have to take the sat in 10th grade ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like .
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would sat ii increase my chances of getting into a good college ?
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let 's see if we can now give ourselves a more tangible understanding of what 's on the new sat and how it might be different than sats of the past . so right over here i have a bullet point list of kind of the major features , the defining features of the new sat , and starting up here are relevant words in context . and there 's two important parts to that , relevant , and context . and when we 're talking about relevant words , these are words that you will actually see when you go to college , or that you will use in your working life . people sometimes say something like , `` hey , that 's an sat word . '' sometimes referring to a word that , maybe , it 's just a hard word that is not so practical . what 's different about the new sat , is that they 're going to be relevant words , and not just relevant words in a vacuum , they 're going to be in a context , in a situation where you might be editing a passage or reading something , so it 's going to matter to understand what those words are , and once again , they 're not going to be some kind of crazy words that you only memorize for a standardized test . and this is an example of that , i have a passage of relevant words in context , and this is going to be from the new sat writing section . and the new sat writing section , as you see , we 're in context here , and this little four with the circle around it , this shows that , hey , this part of the passage is referring to question four . and i 'll look at this real fast . as kingman developed as a painter , his works were often compared to paintings by chinese landscape artists dating back to ce 960 , a time when a strong tradition of landscape paining emerged in chinese art . kingman however , and then the default right here is vacated from that tradition in a number of ways . well , vacated once again , is n't a crazy word , it 's a word that you will hear in everyday language , in college and in work , but it sounds a little bit off . and you look at the choices here , evacuated , departed , retired , once again , all words that you will see in college , and you will see in life , at least for this one , and my job here is n't to answer the sat questions , but hey , this feels like he departed from the traditions . so i would have gone with that . but anyway , this is n't about doing the questions , but more seeing how these questions -- what they actually look like . these are n't crazy words that you would only just memorize for a test , these are useful words to know in life , and once again we have a context of when we would use it . we would use it maybe to edit a passage like this . so the other major themes of the new sat , command of evidence . so we 're going to see in the writing , reading , and math sections to be able to look at either , it could be data , and this could even be in the reading section , hey , what kind of data can help us make different types of conclusions , or even thinking about how we might rewrite something based on the data that 's relevant , and this is an example of it right over here . and once again , i 'm not gon na go into this question , there 's other videos we have where we do example problems , but this is interesting , you 're seeing kind of data evidence , and this is n't even the math section . so this is another example of something you might see or you will see on the new sat . and the reason why that 's important , is because you will see this throughout life . it 's not like reading or writing is somehow separated from data . all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that . now the math portion , this is a big deal that the new sat -- classically the sat had been associated with , hey , maybe it 's a little bit trickier or things like that , but the new sat really is focused on the things that you actually learn in school . so learn in school , and it 's even a subset of that . it 's kind of the most important subset especially for being college and life ready . so the sections or the types of problems you 'll see are the heart of algebra . this is really the core of algebra that really matters . then you 're gon na see the passport to advanced math , and these are the skills that allow you to have the knowledge to get higher level math that you will take in college . so you 're ready to take those pre-calculus , calculus , statistics courses that you might see in college , and then there 's problem solving and data analysis . once again , these are real world types of data analysis things , things that you will see throughout college and in your entire life , so it 's focused on things you learn in school , and things that are going to matter in college and life . now another big theme , and this is across all the sections , is real world contexts . these are n't just tricky problems in some type of vacuum , these are going to be things that you 're going to be seeing . i know i keep repeating that , but you see that theme that they 're not just kind of brain teaser type things , these are things that you will see as you go through your college career . analysis in science and social studies . so even though the sections are formerly reading and math and writing , they 're going to use those sections in order to touch on important topics in science and social studies , so you 're definitely going to see a broader coverage of different types of domains in the new sat . and this is one that i found really interesting , this notion of founding documents and great global conversation as being part of the sat . so it 's a really , i would say , good incentive to get familiar with things like to get familiar with things like the gandhi 's quit india speech , or the declaration of independence , or the federalist papers . you 're going to see this type of thing in the new sat in a reading passage or to kind of make some judgement about some things . and then , the last bullet point here , this is just kind of a high level point , and this emphasizes that this new sat really is about what do you know , not having to think about , hey do i guess , do i not guess ? there 's no penalty for wrong answers . in the past , there was a penalty for wrong answers , but now there 's no penalty . so if you feel like you have a sense of something , or really even if you do n't , you should at least try to answer every question that you get to . you should n't get into all the whole gamesmanship of , okay , should i do it , or should i not . so hopefully this gives you a sense of things , i have some other questions here that maybe make things a little bit more tangible , this one right over here , this is would be the passport to advanced math right over here . you have a non-linear system , but this is really useful . this is something that you will see in school , especially when you get to college , and then they have data analysis questions like this which is very valuable as you , once again , go into college , and really almost any field you go into , you 're gon na have to do data analysis or things like this . so hopefully this gives you an overview , and encourages you to dive in , and the best way , always , to prepare for this , is use the khan academy tools . take the practice test , review the questions that you get really wrong or that you 're not sure about , and then do as much deliberate practice as you need and the system will personalize to your needs so that you get hopefully , the best use of your time . have fun .
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all of these things happen together in the real world , and it might not just be data evidence , it could be looking at evidence , an author 's argument , so it could be written evidence of some sort as well . the essay , the optional essay on the new sat is going to be about analyzing a source . so looking at what 's happening , and analyzing where kind of the conclusions are coming from and we can go into more depth in future videos on that .
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what is a narrative essay ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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could someone explain again what is the difference between tan and tan-1 ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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why did sal use tan-1 instead of tan ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien .
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is n't tan-1 the same as when we divide a square root ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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is n't cot is the inverse of tan ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here .
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would the alien died if the mib man shot with rounded degree and not actual supposed degree ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here .
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what is the difference between a right and a left triangle ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse .
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does the majority of the answers in solving mathematical problems for trigonometry while using soh cah toa have to be in degrees ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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can anyone explain exactly what the inverse tangent is ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle .
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can you solve a triangle if all 3 angles are given when its area is also given ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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so what was the point then of him writing tangent inverse of tangent of theta ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here .
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a tangent is a function , right ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent .
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if a=3 what is sin a ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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inverse of tan theta is 1/tan theta ... ... so , 1/tan theta is 1/ ( opposite / adjacent ) now , this will get reciprocal ... ... .. now it 's called adjacent / opposite or cot theta so now , inverse of tan theta is adjacent / opposite now , ca n't inverse of tan theta be simply called as cot theta ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 .
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could someone explain again what is the difference between sin-1 and tan-1 ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 .
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could sal use sin-1 instead of sin ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse .
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if you know the degree of 1 angle , and the length of the opposite angle , can you find the hypotenuse ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees .
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this may just be my calculator but when i take the tan^-1 of 6 i get his answer but if i input it as ( 324/54 ) i get .10510 ... .. any ideas/suggestions ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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when should you use inverse or regular trig ratios ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent .
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is the difference for when you are finding the sides or the angles ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse .
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when do we have to use soh cah toa ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle .
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could you solve the problem above with one side and one angle ( 90 degrees ) ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here .
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for cc geometry , should you keep the calculator at degree mode ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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what is the difference b/w tan ^-1 and cot ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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how do you use inverse of sine and cosine ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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why was the inverse of tan calculated instead of tan ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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is n't the inverse of tangent called cotangent ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here .
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why do we assume that it is a right angled triangle ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition .
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does sal multiply both sides by tan^-1 like we do in linear equations ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript .
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why does sal use the inverse tan function ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse .
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for the question , instead of using soh cah toa , could you not just use the cosine rule that states : 324 squared + 54 squared - ( 2ab ) cos 90 ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ?
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what is an inverse tangent ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 .
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how do i input inverse tangent in calculator ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ?
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how would you find a length given one side ( not hypotenuse ) and three other angles ?
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black . a men in black agent is standing at ground level , 54 meters across the eiffel square . so 54 meters from , i guess you could say the center of the base of the eiffel tower , aiming his laser gun at the alien . so this is him aiming the laser gun . at what angle should the agent shoot his laser gun ? round your answer , if necessary , to two decimal places . so if we construct a right triangle here , and we can . so the height of this right triangle is 324 meters . this width right over here is 54 meters . it is a right triangle . what they 're really asking us is what is this angle right over here . and they 've given us two pieces of information . they gave us the side that is opposite the angle . and they 've given us the side that is adjacent to the angle . so what trig function deals with opposite and adjacent ? and to remind ourselves , we can write , like i always like to do , soh , cah , toa . and these are really by definition . so you just have to know this , and soh cah toa helps us . sine is opposite over hypotenuse . cosine is adjacent over hypotenuse . tangent is opposite over adjacent . we can write that the tangent of theta is equal to the length of the opposite side -- 324 meters -- over the length of the adjacent side -- over 54 meters . now you might say , well , ok , that 's fine . what angle , when i take its tangent , gives me 324/54 ? well , for this , it will probably be useful to use a calculator . and the way that we 'd use a calculator is we would use the inverse tan function . so we could rewrite this as we 're going to take the inverse tangent -- and sometimes it 's written as tangent with this negative 1 superscript . so the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54 . and just to be clear , what is this inverse tangent ? this just literally says , this will return what is the angle that , when i take the tangent of it , gives me 324/54 . this says , what is the angle that , when i take the tangent of it , gives me tangent of theta ? so this right over here , this just simplifies to theta . theta is the angle that when you get the tangent of it gets you tangent of theta . and so we get theta is equal to inverse tangent of 324/54 . once again , this inverse tangent thing you might find confusing . but all this is saying is , over here , we 're saying tangent of some angle is 324/54 . this is just saying my angle is whatever angle i need so that when i take the tangent of it , i get 324/54 . it 's how we will solve for theta . so let 's get our calculator out . and let 's say that we want our answer in degrees . well , i 'm just going to assume that they want our answers in degrees . so let me make sure my calculator is actually in degree mode . so i 'll go to the 2nd mode right over here . and actually it 's in radian mode right now . so let me make sure i 'm in degree mode to get my answer in degrees . now let me exit out of here . and let me just type in the inverse tangent -- so it 's in this yellow color right here -- inverse tangent of 324 divided by 54 is going to be -- and they told us to round to two decimal places -- 80.54 degrees . so theta is equal to 80.54 degrees . that 's the angle at which you should shoot the gun to help defeat this horrible alien .
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a tiny but horrible alien is standing at the top of the eiffel tower -- so this is where the tiny but horrible alien is -- which is 324 meters tall -- and they label that , the height of the eiffel tower -- and threatening to destroy the city of paris . a men in black -- or a men in black agent . i was about to say maybe it should be a man in black .
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what is a men in black agent ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together .
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or more clearly , why would you have to describe the attributes of a dolphin , if in your perspective , it could be anything ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over .
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how does step `` 0 '' exist ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite .
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how come when someone akes a question vi doesent aculay answer it someelse does how come ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite .
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how mathematically pythagorean theorem can be proofed ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together .
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how can we define s.a.s ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same .
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hi vi , how can you fold a square into 3 equal parts ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times .
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i know to figure this out is a2+b2=c2 , but how are you suppose to know which side is a , b , or c ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof .
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how long did it take vi hart to make this video ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease .
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how do you fold a paper 12 times ?
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you do n't need numbers or fancy equations to prove the pythagorean theorem , all you need is a piece of paper . there is a ton of ways to prove it , and people are inventing new ones all the time , but i am going to show you my favorite . only instead of looking at diagrams , we 're gon na fold it . first , you need a square , which you can probably obtain from a rectangle if you ask nicely . step one , fold your square in half one way , then the other way , then across the diagonal . no need to make these creases sharp , we 're just taking advantage of the symetries of the square for the next step . but , be precise . step 2 : make a crease along this triangle , parallel to the side of the triangle that has the edges of the paper . you can make it anywhere you want . this is where you are choosing how long and pointy , or short and fat , your right triangle is going to be , because this is a general proof . now when you unwrap it , you 'll have a square centered in your square . extend those creases and make them sharp , and now we 've got we 've got four lines all the same distance from the edges , which will allow us to make a bunch of right triangles that are all exactly the same . step three : fold from this point to this one . basically taking a diagonal of this rectangle . now we 've got our first right triangle . which has the same shape and area as this one . let 's call the sides : `` a little leg '' , `` a big leg '' , and `` hypothenus '' . rotate ninety degrees , and fold back another triangle , which of course is just like the first . repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles . so the area is the hypotenus squared . step four : unfold , and this time let 's choose a different four triangles to fold back . rip along one little leg , and fold back these two triangles . then you can fold back another two over here . the area of the unfolded paper , minus four triangles , must be the same , no matter which four triangles you remove . so let 's see what we 've got . we can divide this into two squares , this one has sides the length of the little leg of the triangle . and this one has sides as long as the big leg . so the area of both together , is little leg squared , plus big leg squared . which has to be equal to this area , which is hypotenus squared . if you called the sides of your triangle something more abstract , like : a , b , and c , you 'd of course have a squared plus b squared equals c squared so quick review : step 0 : aquire a paper square . ok , step one : fold it in half three times . step 2 : fold parallel to the edges anywhere you choose and extend the crease . step three : fold back four right triangles around the square and admire the area hypothenus squared that is left over . step four : unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left . and that is all there is to it ! of course , mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves . so be sure to not believe me when i tell you things like : this is a square . think of a few ways you could convince yourself that no matter what the triangles on the outside look like , this will always be a square , and not some kind of a rombus or parallelogram or dolphin or something . or , you know , maybe it is a dolphin , in which case you should define what a dolphin is and then show that this fits that definition . also , these edges look like they line up together . do they always do that ? is it exact ?
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repeat on the following two sides . the original paper minus those four triangles , gives us a lovely square . how much paper is this ? well , the length of a side is the hypotenus of one of these triangles .
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hey vi how did your paper suddenly split or did it just trick my eyes ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know .
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if the first condition for linearity is satisfied , wo n't the second condition be satisfied too ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector .
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if we know that the transformation of two added vectors is the same as their transformations added , then multiplying one vector by c could be seen as adding another vector to it also , right ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that .
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is it simply to find the perpendicular line to our equation , or rather the equation of the original equation flipped over the y=x axis ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question .
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or does invertibility imply linearity ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this .
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to add on to fares comment : do n't we only need the fact that t is injective to show that t ( a ) = t ( b ) implies a = b ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there .
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does associativity only apply to linear transformations ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product .
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regarding syntax , t^-1 ( b ) , b , should have a vector arrow above it , and at 1 ax , should also have a vector arrow above the x ?
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i 've got a transformation t. when you apply the transformation t to some x in your domain , it is equivalent to multiplying that x in your domain , or that vector . by the matrix a . and let 's say we know the linear transformation t can be -- that 's transformation matrix when you put it in reduced row echelon form -- it is equal to an n by n identity matrix , or the n by n identity matrix . well this alone tells a lot of things . first of all , if when you put this in reduced row echelon form , you get a square identity matrix . that tells us that the original matrix had to be n by n. and it also tells us that t is a mapping from rn to rn . and we saw in the last video , all of these are conditions , especially this one right here , for t to be invertible . so if we know that this is true -- t is a linear transformation , it 's a reduced row echelon form of the transformation matrix of the identity matrix right there -- we know that t is invertible . let 's remind ourselves what it even means to be invertible . to be invertible means that there exists some -- we used the word function before , but now we 're talking about transformations , they 're really the same thing . so let 's say there exists some transformation -- let 's call it t-inverse like that , or t to the minus 1 -- such that the composition of t-inverse with t is equal to the identity transformation on your domain , and the composition of t with t-inverse is equal to the identity transformation on your codomain . just like that . and just to remind you what this looks like , let 's draw our domains and codomains . our domain is rn , and our codomain is also rn . so if you take some vector in your domain , apply the transformation t -- you 're going to go into your codomain , so that is t. and then if you apply the t-inverse after that , you 're going to go back to that original x . so this says , look , you apply t and then you apply t-inverse . you 're just going to get back to where you started . it 's equivalent to the identity transformation . just like that . this is saying if you start here in your codomain , you apply this inverse transformation first , then you apply your transformation , you 're going to go back to the same point in your codomain . so it 's equivalent to the identity transformation in your codomain . it just happens to be in this case that the domain and the codomain are the same set , rn . now we know what a transformation -- what it means to be invertible . we know what the conditions are for invertibility . so this begs the next question . we know this guy is a linear transformation , in fact that 's one of the conditions to be able to represent it as a matrix . or any transformation that can be represented as a matrix vector product is a linear transformation . so this guy 's a linear transformation . but the question is , is t-inverse a linear transformation ? let 's review what the two conditions are that we need to have to be a linear transformation . so we know t is a linear transformation . so we know that if you apply the transformation t to two vectors -- let 's say x and y -- if we apply to the sum of those two vectors , it is equal to the transformation of the first vector plus the transformation of the second vector . that 's one of the conditions , or one thing that we know is true for all linear transformations . and the second thing we know is true for all linear transformations is , if we take the transformation of some scaled version of a vector in our domain , it is equal to the scaling factor times the transformation of the vector itself . these are both conditions for linear transformations . so let 's see if we can prove that both of these conditions hold for t-inverse , or this guy right here . so to do this , let 's do this little exercise right here . let 's take the composition of t with t-inverse of two vectors , a plus b . remember , t-inverse is a mapping from your codomain to your domain , although they 're both going to be rn in this case . but t-inverse maps from this set to that set . let 's write it up here . t-inverse is a mapping from your codomain to your domain . although it looks identical , just like that . ok , so what is this going to be equal to ? well we just said , by definition of your inverse transformation , this is going to be equal to the identity transformation on your codomain . so assuming these guys are members of your codomain , in this case rn , this is just going to be equal to a plus b . this thing , the composition of t with its inverse , by definition is just the identity transformation on your codomain . so this is just whatever i put in here . if i put in an x here , this would be an x . if i put in an apple here , this would be an apple . it 's going to be the identity transformation . now what is this equal to ? well i could use the same argument to say that this right here is equal to the identity transformation applied to a . and i 'm not writing the identity transformation , i 'm writing this . but we know that this is equivalent to the identity transformation . so we could say that is equivalent to the composition of t with the inverse applied to a , and we could say that this is the equivalent to the identity transformation , which we know is the same thing as t , the composition of t with t-inverse applied to b . so we can rewrite this thing right here as being equal to the sum of these two things . in fact we do n't even have to rewrite it . we can just write it 's equal to -- this transformation is equal to this . and maybe an easier way for you to process it is , we could write this as t of the t-inverse of a plus b , is equal to t of the t-inverse of a plus t of the t-inverse of b . and this should -- i do n't know which one your brain processes easier , but either of these , when you when you take the composition of t with t-inverse , you 're going to be left with an a plus b . you take the composition of t with t-inverse , you 're left with an a . you take the composition of t with t-inverse , you 're just left with a b there . so in either case you get a plus b -- when you evaluate either side of this expression you 'll get the vector a plus the vector b . now what can we do ? we know that t itself is a linear transformation . and since t is a linear transformation , we know that t applied to the sum of two vectors is equal to t applied to each of those vectors and summed up . or we can take it the other way . t applied to two separate vectors -- so we call this one vector right here , and this vector right here . so in this case i have a t applied to one vector , and i 'm summing it to a t applied to another vector . so it 's this right here , which we know is equal to t applied to the sum of those two vectors . so this is t applied to the vector t-inverse of a -- let me write it here -- plus t-inverse of b . it might look a little convoluted , but all i 'm saying is , this looks just like this . if you say that x is equal to t-inverse of a , and if you say that y is equal to t-inverse of b . so this looks just like that . it 's going to be equal to the transformation t applied to the sum of those two vectors . so it 's going to equal the transformation t applied to the inverse of a plus t-inverse of b. i just use the fact that t is linear to get here . now what can i do ? let me let me simplify everything that i 've written right here . so i now have -- let me rewrite this . this thing up here , which is the same thing as this . t , the composition of t , with t-inverse , applied to a plus b is equal to the composition -- or actually not the composition , just t -- applied to two vectors , t-inverse of a plus t-inverse of vector b . that 's what we 've gotten so far . now we 're very close to proving that this condition is true for t-inverse , if we can just get rid of these t 's . well the best way to get rid of those t 's is to take the composition with t-inverse on both sides . or take the t-inverse transformation of both sides of this equation . so let 's do that . so let 's take t-inverse of this side , t-inverse of that side , should be equal to t-inverse of this side . because these two things are the same thing . so if you put the same thing into a function , you should get the same value on both sides . so what is this thing on the left-hand side ? what is this ? this is the composition -- let me write it this way -- this is the composition of t-inverse with t , that part , applied to this thing right here . i 'm just changing the associativity of this -- applied to t-inverse of the vector a plus the vector b . that 's what this left hand side is . this part right here , t-inverse of t of this , these first two steps i 'm just writing as a composition of t-inverse with t applied to this right here . that right there is the same thing as that right there . so that was another way to write that . and so that is going to be equal to the composition of t-inverse with t -- i 'll write that in the same color -- a composition of t-inverse with t. that 's this part right here , which is very similar to that part right there -- of this stuff right here , of t-inverse of a plus t-inverse of the vector b . now by definition of what t-inverse is , what is this ? this is the identity transformation on our domain . this is the identity transformation on rn . this is also the identity transformation on rn . so if you apply the identity transformation to anything , you 're just going to get anything . so this is going to be equal to -- i 'll do it on both sides of the equation -- this whole expression on the left-hand side is going to simplify to the t-inverse of the vectors a plus the vector b . and the right-hand side is just going to simplify to this thing . is equal to -- because this is just the identity transformation -- so it 's just equal this one , t-inverse of the vector a plus t-inverse of the vector b . and just like that , t-inverse has met its first condition for being a linear transformation . now let 's see if we can do the second condition . let 's do the same type of thing . let 's take the composition of t with t-inverse , let 's take the composition of that on some vector , let 's call it ca . just like that . well we know what this is equal to , this is equal to the identity transformation on rn . so this is just going to be equal to ca . now what is a equal to ? what is this thing right there -- i 'll write it on the side right here , let me do it in an appropriate color . or we could say that a , the vector a is equal to the transformation t with the composition of t with t-inverse applied to the vector a . because this is just the identity transformation . so we can rewrite this expression here as being equal to c times the composition of t with t-inverse applied to my vector a . and maybe it might be nice to rewrite it in this form instead of this composition form . so this left expression we can just write as t of the t-inverse of c times the vector a -- all i did is rewrite this left-hand side this way -- is equal to this green thing right here . well i 'll rewrite similarly . this is equal to c times the transformation t applied to the transformation t-inverse applied to a . this is by definition what composition means . now t is a linear transformation . which means that if you take c time t times some vector , that is equivalent to t times c times t applied to c times that vector . this is one of the conditions in your transformation . so this is always going to be the case with t. so if this is some vector that t is applying to , this is some scalar . so this thing , because we know that t is a linear transformation , we can rewrite as being equal to t applied to the scalar c times t-inverse applied to a . and now what can we do ? well let 's apply the t-inverse transformation to both sides of this . let me rewrite it . on this side we get t of t-inverse of ca is equal to t of c times t-inverse times a . that 's what we have so far . but would n't it be nice if we could get rid of these outer t 's ? and the best way to do that is to take the t-inverse transformation of both sides . so let 's do that . t-inverse -- let 's take that of both sides of this equation , t-inverse of both sides . and another way that this could be written . this is equal into the composition of t-inverse with t applied to t-inverse applied to c times our vector a . this right here , i just decided to keep writing it in this form , and i took these two guys out and i wrote them as a composition . and this on the right-hand side , you can do something very similar . you could say that this is equal to the composition of t-inverse with t times -- or not times , let me be very careful . taking this composition , this transformation , and then taking that transformation on c times the inverse transformation applied to a . let me be very clear what i did here . this thing right here is this thing right here . this thing right here is this thing right here . and i just rewrote this composition this way . and the reason why i did this is because we know this is just the identity transformation on rn , and this is just the identity transformation on rn . so the identity transformation applied to anything is just that anything . so this equation simplifies to the in t-inverse applied to c times some vector a , is equal to this thing , c times t-inverse times some vector a . and just like that , we 've met our second condition for being a linear transformation . the first condition was met up here . so now we know . and in both cases , we use the fact that t was a linear transformation to get to the result for t-inverse . so now we know that if t is a linear transformation , and t is invertible , then t-inverse is also a linear transformation . which might seem like a little nice thing to know , but that 's actually a big thing to know . because now we know that t-inverse can be represented as a matrix vector product . that means that t-inverse applied to some vector x could be represented as the product of some matrix times x . and what we 're going to do is , we 're going to call that matrix the matrix a-inverse . and i have n't defined as well how do you construct this a-inverse matrix , but we know that it exists . we know this exists now , because t is a linear transformation . and we can take it even a step further . we know by the definition of invertibility that the composition of t-inverse with t is equal to the identity transformation on rn . well what is a composition ? we know that t , if we take -- let me put it this way . we know that t of x is equal to ax . so if we write t-inverse , the composition of t-inverse with t applied to some vector x is going to be equal to first , a being applied to x is going to be equal to ax , this a right here , ax . and then you 're going to apply a inverse-x , you 're going to apply this right here . and we got that this is the equivalent to -- when you take the composition , it 's equivalent to , or your resulting transformation matrix of two composition transformations is equal to this matrix matrix product . we got that a long time ago . in fact that was the motivation for how a matrix matrix product was defined . but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again . so we get a very interesting result . a-inverse times a has to be equal to the identity matrix . a-inverse , or the matrix transformation for t-inverse , when you multiply that with the matrix transformation for t , you 're going to get the identity matrix . and the argument actually holds both ways . so we know this is true , but the other definition of an inverse , or invertibility , told us that the composition of t with t-inverse is equal to the identity transformation in our codomain , which is also rn . irn . so by the exact same argument , we know that when you go the other way , if you apply t-inverse first and then you apply t -- so that 's equivalent of saying apply t-inverse first , and then you apply t to some x vector , that 's equivalent to multiplying that x vector by the identity matrix , the n by n identity matrix . or you could say , you could switch the order . a times a-inverse is also equal to the identity matrix . which is neat , because we learned that matrix matrix products , when you switch the order they do n't normally always equal each other . but in the case of an invertible matrix and its inverse , order does n't matter . you can take a-inverse times a and get the identity matrix , or you could take a times a-inverse and get the identity matrix . now we 've gotten this far , the next step is to actually figure out how do you construct . we know that this thing exists , we know that the inverse is a linear transformation , that this matrix exists . we see this nice property , that when you multiply it times the transformation matrix you get the identity matrix . the next step is to actually figure out how to figure this guy out .
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but what 's interesting here is , this composition is equal to that , but it 's also equal to the identity transformation on rn applied to that vector x , which is equal to the identity matrix applied to x . right ? that is the n by n matrix , so when you multiply by anything , you get that anything again .
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but we do n't know for sure that a and b are members of the domain of t-1 right ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 .
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would we multiply the final result shown by 4 ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 .
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would you be able to simplify the factorial expression by dividing by 9 ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 .
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how would you compute the number of distinct permutations you can get from the word architecture ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
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always resulting in an integer ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order .
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how would the formula for combination and permutation change if k > n ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 .
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what if the denominator became 0 factorial ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 .
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why does sal put the denominator in parenthesis when he 's using the calculator ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many 6-card hands are possible ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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fair enough . how many 9 card hands are possible ? so let 's think about it .
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how many fewer 5-card hands are available than 6-card hands ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this .
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so i wondered : when taking 8 random questions from a pool of 20 possible questions , what are the odds of having 2 or more times the same question ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand .
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a card is drown at random from a well suffel packs of cards.what is the probability that it is a heart or a quine ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot .
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how many unique ways are there to arrange the letters in the word that ?
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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are possible ? so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them . so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ? i 'm going to pick nine cards for my hand . and so for the very first card , how many possible cards can i pick from ? well , there 's 36 unique cards , so for that first slot , there 's 36 . but then that 's now part of my hand . now for the second slot , how many will there be left to pick from ? well , i 've already picked one , so there will only be 35 to pick from . and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here . maybe i have a -- let me put it here -- maybe i have a 9 of spades here , and then i have a bunch of cards . and maybe i have -- and that 's one hand . and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards . or maybe another hand is i have the eight cards , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and then i have the 9 of spades . if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't matter . so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot . then in the second slot , i have 8 ways of putting a card in the second slot , because i took one to put it in the first , so i have 8 left . then 7 , then 6 , then 5 , then 4 , then 3 , then 2 , then 1 . that last slot , there 's only going to be 1 card left to put in it . so this number right here , where you take 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 , or 9 -- you start with 9 and then you multiply it by every number less than 9 . every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we want to divide by the number of ways we can order things so that we do n't overcount . and this will be an answer and this will be the correct answer . now this is a super , super duper large number . let 's figure out how large of a number this is . we have 36 -- let me scroll to the left a little bit -- 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 , divided by 9 . well , i can do it this way . i can put a parentheses -- divided by parentheses , 9 times 8 , times 7 , times 6 , times 5 , times 4 , times 3 , times 2 , times 1 . now , hopefully the calculator can handle this . and it gave us this number , 94,143,280 . let me put this on the side , so i can read it . so this number right here gives us 94,143,280 . so that 's the answer for this problem . that there are 94,143,280 possible 9 card hands in this situation . now , we kind of just worked through it . we reasoned our way through it . there is a formula for this that does essentially the exact same thing . and the way that people denote this formula is to say , look , we have 36 things and we are going to choose 9 of them . right ? and we do n't care about order , so sometimes it 'll be written as n choose k. let me write it this way . so what did we do here ? we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 . so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 . so 27 factorial -- so let 's think about this -- 36 factorial , it 'd be 36 times 35 , you keep going all the way , times 28 times 27 , going all the way down to 1 . that is 36 factorial . now what is 36 minus 9 factorial , that 's 27 factorial . so if you divide by 27 factorial , 27 factorial is 27 times 26 , all the way down to 1 . well , this and this are the exact same thing . this is 27 times 26 , so that and that would cancel out . so if you do 36 divided by 36 , minus 9 factorial , you just get the first , the largest nine terms of 36 factorial , which is exactly what we have over there . so that is that . and then we divided it by 9 factorial . and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find out all of the possible ways you can pick k things from those n things , and you do n't care about the order . all you care is about which k things you picked , you do n't care about the order in which you picked those k things . so that 's what we did here .
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so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ?
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would n't it be 36 factorial over nine factorial instead of 36*35*34 ... 30*29*28 ?
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