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Trigonometry: Using a Cofunction Identity


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:27
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 102 MB
  • Posted: 07/01/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Trigonometric Identities (23 lessons, $26.73)
Trigonometry: Other Advanced Identities (3 lessons, $4.95)

Taught by Professor Edward Burger, this lesson comes from a comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers trigonometric functions, trigonometric identities, application of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, and more.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Using a Cofunction Identity
Now we can use these functions, these formulas for sums and difference of angles, actually to produce new identities. Sometimes these identities are referred to as cofunctions, because they're sort of two functions that are related to each other. Let's start off with the following. Let's see if you could figure out what equals.
First of all, let's make a guess, looking at the graph. So when you think of the graph of cosines, it looks like this. Now what does it mean to replace x by ? Remember, "add to x, go west." So since I'm subtracting, I should go east. So I should move this whole picture to the east by . So what happens exactly? Well, you have to visualize this in your mind's eye. If I slide this over, I get something that looks like this. I get the sine function. So in fact this is the sine function. Let's see if that's really right. Let's see if we can confirm that punch to see if that's really okay.
Well, what do I do? Well, just take a look at and use the difference formula for cosine and see what this would give me. Well, okay. Remember cosine is evil. It's the bad seed, so it wants all the credit for itself. So it's going to be multiplied by . In fact, it's so evil that since you're subtracting here, it's going to add, and then it throws in all the signs. Since it has to be there anyway, I'll throw it in.
All right. Well, what is this? Well, we have times . But look what is; is zero. So in fact, this term drops out, so I get a zero plus--and what's ? Well, remember that is actually 1. So in fact, this is just 1. So I see . So now we've actually proved, using the difference angle formula for a cosine, that if you take cosine and shift it to the right radiant, you actually get the sine function. Now you know it exactly. So before it looked that way when you did all that shifting but we weren't sure if the curves matched up just perfectly. But now for the first time we're sure. Everything matches up just perfectly because of this great formula we can use.
Let's try another one. Let's verify what happens if we take . So what happens if we shift the sine function--this is going to be slightly hard to visualize. I'm going to take the sine function and shift it over, so what would it look like? Well, then it would sort of start down here and go up and come down like this. You see I'm just shifting this over so it sort of has this wave that's been displaced a little bit. What function is that? Well, if you think about it, it's not cosine but it's the negative of cosine. Cosine does this, and this function would sort of do the negative of it. So it would just be the flip of cosine. If I were able to flip this, I would get that function. So it should be negative cosine. Let's see if we can verify that this is really negative cosine.
Okay, so I'm looking at sine of a difference, so I use the difference formula for sine. Sine is a very happy and welcoming function. So we take and we multiply it by . Everyone gets top billing. Since I'm subtracting here, I'll subtract here to keep it fair. And then I'm going to have because I reversed those roles, times . Let's see what we get here. So what is ? Well, is zero, so in fact this term just drops out. And what is ? Well, is 1. So this term drops out and this becomes just 1, so I'm left with that negative sign . And that verifies the formula as we thought. So that's pretty cool. We can verify these things using the sums and differences formula.
Let's try one last one. Let's verify that . Let's see what happens now. So looking at , what do you do there? Well, there I've got . I divide that by 1 and then I switch the sign, plus . That's what that looks like. Now we have to figure out what is. So what's ? Well, let's graph this. So, let's see. You know, I'm getting a little nervous here. In fact, I'm getting really nervous here. I'm getting really nervous because it's not defined. So what in the heck are we thinking here? It's not defined. If you take this thing and you were to shift it over, I admit that you're getting the right thing, but how in the heck do you do this? Okay, let's cut this.
Stop, stop, stop! Before we cut, I just want to show you how you can sort of get around this little problem, and that's why I was stopping. See, the stopping thing is the following. You see, I can't evaluate . It's undefined there. So that's sort of a problem. What do you do with that? Well, I'll show you how we get around that a little bit. I call this sort of dubious math, but it's okay. It's okay. See, this is undefined, so how do you sort of manipulate that? Well, let me perform a little trick here just to show you how we can really get this formula.
Let's just multiple top and bottom by . So I multiply by something that looks like something over itself. It sort of looks like 1. This is a little bit dubious because, of course, that's not even defined, so it's a little bit shaky here. But just play along. This is just sort of a shaky thing, so maybe I should be shaking! But when I do that, just watch formally what happens. When I distribute this, what do I see? I see minus--and then is just 1. And then what do I have on the bottom? I have 1 times that, so that's , and then I have plus--and then there's a cancellation here. I'm just left with . Okay. So that's the dubious part.
Now, what is ? Well, remember is the same thing as cotangent. So in fact, I could rewrite this as . Now if you allow me to do that, which was just sort of acting formally here--very formal, I should be wearing a tuxedo maybe--but I'm just sort of manipulating symbols, not thinking about if it's allowable or not, well now I'm in fine shape. Because what do I know about the cotangent? The cotangent is zero wherever I have an asymptote on the tangent. So in fact now I see that at , the cotangent is zero. So if the cotangent is zero, look what happens. This term is zero because there's a factor here, so this drops out. And this is zero. And so what am I left with? I'm left with . So I see that this thing equals . And is the same thing as . And so I actually did in fact prove this identity after all, but I had to do a little bit of trickery that, hmm, I don't know. But anyway, you can sort of see it formally. And in fact, you can make all of that mathematics really okay and justify it if you're just a little bit more careful. But why bother? You can sort of see, at least formally, this formula holds. And in fact graphically, if you look at the graph of these things and shifts, you'll see the exact formula holds. So graphically we can verify what we can prove here algebraically, but it's a little bit shaky.
Anyway, now you can see that they are all cofunctions. This is a cofunction of that because they're just shifts of each other once you shift a little bit. Anyway, okay. Enjoy these and I hope you're not going to be too shaky yourself. See you soon.

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