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Calculus: Evaluating Inverse Trig Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 100 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Inverse and Hyperbolic Functions (14 lessons, $19.80)
Calculus: Inverse Trigonometric Functions (3 lessons, $4.95)

In this lesson, you will learn how to evaluate inverse trig functions by hand or with a graphing calculator such that you can solve equations that include these functions. The video will also teach you about the special behavior of inverse trigonometric functions and equations with inverse trig functions. To evaluate and solve inverse trig expressions, you will learn to convert them to normal trig expressions. Given that inverse trig functions are only applicable for a restricted domain, they behave differently than most inverses.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

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.

About this Author

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Elementary Functions and Their Inverses
Inverse Trigonometric Functions
Evaluating Inverse Trigonmetric Functions Page [1 of 2]
I thought it would be fun just to take a minute or two to think about these inverse trig functions and actually see how we can evaluate these inverse trig functions when you actually plug in particular numbers. So let’s take a look at some examples together, just to get a sense of this. Really, there’s nothing new here, but it requires us to think backwards, because if you're thinking about an inverse function, you're sort of switching x and y. So it’s sort of like a backward thing. So here we go.
Let’s take a look at the first one. Let’s find arcsin (½). So there’s the question. What is arcsin (½)? Well, here’s how I think about it. I say, “Well, okay, it’s some question mark. Who knows what it is?” and I just convert arcsin. So that means that question mark is the angle whose sine is ½. So the answer, whatever the answer is, that’s the angle whose sine is ½. Well, there are a lot of angles that have sine ½, but on that restricted region, there’s only one, and that’s what we used to call 30 degrees, but now, using radian notation, it’s . So is the angle that’s in the right region, for which the sine of that angle is ½.
Okay, let’s try another one; arctan (1). So this is saying that whatever this is, that’s the angle whose tangent is 1. So what’s the measure of angle, the angle for which tangent is 1? Well, the answer is there are a lot of them, but within that confined region for tangent, for which we’re taking the inverse of, we know that the only possibility would be 45 degrees or, as we like to call it here on the radian ranch, . Now, let me just show you. In fact, you can verify this on a calculator. In fact, on a calculator, depending upon how big the keys are, they might say arcsin or arctan. In fact, what they may say is something like mentioned before, it would be this key, sin-1. I’ve got a calculator right here, let me see how they did it. In fact, they have nothing here, so they just have literally that. And that means the arcsin and so forth. So, for example, if you want to find the arctan (1), let me show you how you do that right now. So what you do is you take the calculator and you type in arctan, so the tangent inverse, and you plug in 1 and you close the little parentheses, and then you write equals, and you get this answer. You get this answer 78, blah, blah, blah. Now, the answer I claimed was . Is that the same answer? Well, let’s actually compute ? divided by 4 and see what happens. So I’ll take ? and I’ll divide it by 4 and look what happens; you get the exact same answer. So, in fact, the answer the computer gets is the numeric version of . So you see how that works? That’s pretty cool and not that hard, it’s just literally thinking of things backwards.
So let’s try another one together; arcsin of – and I’m going to do something sneaky here – (sin(?)). So this is arcsin (sin(?)). A lot of “of’s” here. Let’s think about this and sort of break this down, deconstruct it if you’re one of these New Age philosophers. Everyone’s deconstructing everything. Let’s construct. Let’s build something together. Let’s not deconstruct it, let’s put it back together. Okay, so what is sin (?)? sin (?), if you think about the sine function – I’ll do it for you live, here’s the sine function. So where’s ?? ? stopped right there. So it’s zero. So sin (?) = 0. So now I'm looking for the angle whose sine is zero. But on that restricted interval, where we’re defining the inverse function, the answer is zero radiants. Zero radiants is the angle, for which the sin (0) = 0. So the answer is zero. And this brings up an important and annoyingly good point, because we’re following along this really carefully. We know that this and this, these two are inverse functions. So shouldn’t it be the case that if it’s the inverse functions, shouldn’t they cancel each other out and I’m just left with ?? It seems like a problem. So what’s wrong? Well, what’s wrong is nothing. It’s absolutely perfect, but you have to remember that, with these trigonometric functions, they're only defined on very specific regions. And so the moral here, the life lesson that I want you to take home with you is that it’s not the case that arcsin (sin (x)) = x. So this is not the case. This is not always the case, and here’s an example. It will be the case when you only use x’s that are in the allowable region, the allowable domain. Then, of course, it’s a sine. But if you're going to shift off somewhere, the first thing I’ve got to do is say, “Okay, I've got to shift it back, so the answer’s going to be different.” So, in fact, this is not a surprise, but something that we need to realize is permissible and can happen, since we’re only looking at this small window. Remember we blocked off everything and we’re only looking at these trig functions along this very narrow band. Now, you might be saying, “Ed, you're a wacko. Maybe you're wrong.” So let’s see what the calculator says, as if you can believe the calculator. Why would you believe the calculator over me? In fact, I’m almost offended that you even thought that, that you would trust the calculator, some piece of thing, and here I am telling – all right fine, fine, you don’t trust me, that’s fine.
So I have sin-1, arcsin, then I put in sin, and then I’ll put in ?, just like you wanted. That’s okay, I’m not offended. Well, I am offended, but what do you care? And I claimed it was zero. Let’s see what the calculator says. Here you go, take a look. There you go, you’ve got it. So really, we have to be very careful and remember that on these restricted domains, peculiar things can happen. Even though we know in our hearts that these are inverse functions, they only make sense and they only have this property when you’re looking up on the appropriate domain. So be really careful about that.
Okay, anyway, what about if we were to actually have an equation that involved these sort of arctan, arccos stuff? How would you deal with that? Well, actually, it’s exactly the same kind of thing that we were doing in evaluating these values here. The only difference is that we’d have variables stuck in there. So let’s just try one and see how that would go.
So, for example, suppose that we took a look at arctan of – and instead of putting a number in here, let me just put in some sort of really complicated–looking thing; arctan (2x – 3) = . Now suppose I wanted to find the x’s or the x that makes this thing true. How would you do that? Well now, instead of typing into a calculator or something, we actually have to use some algebra. But the trick or the secret is always the same, to, first of all, admit that we don’t like arctan. Instead, we like the more familiar tangent. So the question is how can we convert this to a tangent thing? Well, tangent is the inverse function of arctan, so let’s think about what this means. This says = the angle whose tangent is that. So, actually I could rewrite this statement in an equivalent way, which doesn’t use arctan, but instead uses tan. The only thing is I have to reverse the roles of these thing, switch x and y. So again, let’s think of this together. So this is saying = the angle whose tangent is 2x – 3. So an equivalent statement would be tan = 2x – 3, because is the angle whose tangent is 2x – 3. So these two things are actually the identical statement, but now this is a happy fact, because I can think about this. tan , that’s one of the popular angles where I actually know what the tangent is. The tangent there is 1. And so, look, I reduced this to just 1 = 2x – 3, and I can solve this. I bring the 3 over to this side, it becomes 4 = 2x, and I see that x = 2. So that’s the answer. And, in fact, you can actually check to make sure. Take the 2 and plug it back in here and I see 2 × 2 = 4 – 3 = 1. And so I’m looking for the angle whose tangent is 1. And that’s . So it checks.
So anyway, now you know for sure everything there is to know about inverse trig functions. I’ll see you at the next lecture.

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