Calculus: Inverse Sine, Cosine, and Tangent

Clear, consise, excellent.

01/26/2009

~ travis5818

That's about it. Ed is solid as usual. I only wished he did the entire book, lol.

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Elementary Functions and Their Inverses
Inverse Trigonometric Functions
The Inverse Sine, Cosine and Tangent Functions Page [1 of 2]
Okay, so let’s take a look at the inverse functions for the standard trigonometric functions. Now, first of all, the question that we have to ask is, “Do trigonometric functions have inverses?” Well, there’s one way to find out. We could look at the graph and ask if the function is always increasing or always decreasing or something like that. So here I just happen to have the graph of sin x, ever popular and much beloved. And you can see that actually it fails the horizontal line test to see if the function is one-to-one, which means that for every y value there’s, at most, one x value that got mapped to it. And, in fact if you think about the sine curve going on forever, this thing is infinite:1, because every time you stop anywhere between –1 and 1, it’s going to hit the curve infinitely many times, because it keeps cycling back and forth. So this is as far from being one-to-one as you can imagine. You can see that by the fact you go up into this valley to a hill, and then you go down the valley and hill, valley and hill, valley and hill forever. This is not a one-to-one function. This function does not have an inverse.
Now, that’s sad, because you like to have inverses for trigonometric functions, because how else can we as teachers torture Calculus II students? Well, the answer is that we can do it, but what we’ll to do is not look at the whole picture. You know, when they say you’ve got to look at the big picture? Well, they’re wrong in this case. In fact, in this case, you have to look at the small picture. Let’s think about it. What if, instead of looking at the entire sine, we only looked at a little piece of it? For example, suppose we just looked at that little piece right there. If I just ignored everything else and looked at that piece, that, in fact, is an increasing function. So that has an inverse. And if you want to find the inverse, what do you do? You just flip along this line here and you would see a little piece of it over here, I guess. So let me ask this question – well, that’s sort of a modest piece, maybe I can actually have a larger piece. In fact, I could have that, that’s still increasing. So what’s the biggest region where I am one-to-one or the biggest region where I am just increasing? Well, let’s try it and see. So if I move this way, I get to , which we know has the high point. That’s the maximum. If I keep going, notice now that I dip, I now have come down. So that’s no good. I failed the horizontal line test, so I’d have to stop at . So I stop at , but maybe I could move a little bit more this way. Let’s see what happens if I move this way. Oh look! This is fine; this is fine. I’m at , but after that, it starts to swing up, so that’s no good. So the biggest interval, the biggest region would be between and . If I go any further on either side, it will no longer be one-to-one. It’ll backup onto itself.
Now, you might say, “He, Ed, wait a minute. Is that the only place you could look? What if we looked here? It’s all one-to-one here; it’s just decreasing. Couldn’t we use that?” And the answer is absolutely. You could use that if you wanted to, too. In fact, you can see there are infinitely many places where the things are just rising up at this spot, and then this spot, and this goes on. And they're infinitely many places where the thing is coming down. You could use any of those intervals to deem as the function that we’ll take the inverse of. So we need to make a convention. We have to have a convention. In fact, that’s what mathematicians do; they have conventions. They all get together and they sit around and they sort of stare at each other. And the convention that they made, in this case, for sine function is let’s just look sort of symmetrically from to , and we’ll just look at the sine function there. We’re going to just restrict the domain. We’re going to restrict the allowable x values from here to here. And if I do that, then if I take the mirror and set it up, we can see what the inverse function would have to look like there. It would have to look something like this. Do you see how that’s a reflection of this line? Look how symmetric that looks over the line. Do you see it? So this curve here would be the inverse of the sine curve. Now, what do we call that? Well, we could call it sin-1, and some people like to do that. I don’t, because sometimes I get it confused with the one over thing, so what I tend to do is write it another way, which is to write it as arcsin. So arcsin is just a fancy way, in this case, of specifying the inverse of the sine function. And you know how I think about this? Arcsin. You could think of as the angle whose sine = x. Because remember if this is the inverse function and I have y = x, then the inverse function would be to flip the x and the y, which would mean that y = arcsin x. That means that I’ve switched the roles of this. x = sin y. y = arcsin x.
So what does the graph of that look like? Well, the graph of that looks like this. There it is. Now, you’ll notice, for example, that it only goes from top to bottom, from to . It’s sort of a funny-looking function, this arcsin x. It just starts here and ends here. It doesn't go on forever, because remember we had a stop, the sine function, in order to make sure it was one-to-one, in order to make sure that we can actually find the inverse. So, in fact, for arcsin, y = arcsin x, the inverse of the sine function, that’s just going to look like this piece right here. That’s the graph of it. That’s all there is. And you can notice that the y’s just range between and , and the x’s, the domain, are only from –1 to 1. There’s nothing to find out here or out here, nothing to find up here or down here. It just lives in this teeny area and it’s that little piece. It’s that little snippet of the sine curve, where we can declare we’ll look at the inverse for it. So there we can make a function one-to-one by restricting the domain.
Okay, what about cosine? Well now, once you see this idea, you can sort of see how to move. Here’s the cosine function, and so now the question is how can we figure out the angle whose cosine is x or the inverse function? And so here we can do the same thing. We just restrict the domain. So how do we restrict the domain? Well, I’ll tell you. There’s a lot of ways of doing it. If I sort of start here and go down, I can go all the way to ?. If I keep going, it backs up into itself, so that’s too far. So here is the largest region where I’ve got a complete sort of decreasing area. Now, what can I do there? Well, there I could now imagine taking the inverse, flipping over the y = x line, and what would I see? I would see this picture right here, and let me try to show you that with my mirror here. All you can see is that little piece right there, which corresponds to this piece right here. So you can just see that little teeny piece right there in the reflection. It’s hard to point to a reflection, by the way, but there it is. If you reflect this whole line and you’ll see this negative part would get reflected to the negative area here, and that’s what you're seeing here. And notice again the restrictions. It’s a very funny function. y only goes from zero to ?. And the x’s only go from –1 to 1. That’s the whole picture. It’s not like I’ve cut anything off. That is a complete picture. The cosine function, the beat goes on. It goes on forever. But if you wanted to find the inverse, we just look here. And again you may say, “Hey, wait a minute, Ed. Suppose I want to look right over there.” Well, you could actually then declare your own special inverse function of cosine right along here, and that’s fine. You won’t be able to talk to any mathematicians, because everyone else was at the convention and agreed with this. But everything will be fine, except that you’d be off a little bit. So let’s just all adopt the convention that we’re only going to live between zero and ?, in which case the inverse function would look like this. If you flipped this picture, you’d get that. So aren’t conventions great?
All right, how about tangent? The tangent’s sort of a crazy function. The tangent goes on, in fact, you can’t see all the tangent, but there’s more stuff even up here. Now tangent, if I just go from these asymptotes, to , that’s just always going up. And if I add any little bit, that’s no good. It now fails the horizontal line test. You see how I hit at two points there? So I can’t go that way and I can’t go this way either for the same reason. So I just have to live between any two consecutive asymptote people. And what do you do there? Well, there if you imagine flipping over the y = x line, this part would bow out and it would come this way, and this would come down and go that way. And so the inverse function would look like this, and this is y = arctan, or y = arctan x. And you can see it’s just this picture flipped along this line. We’ll put this axis back in so you can really sort of see that if you flip it, this thing goes up there – and notice I have these asymptotes, these horizontal asymptotes at and . Those are now horizontal, they used to be vertical, but if you flip, they become horizontal. And you’ll notice that the function here now is defined for all x, but the y’s are restricted between and .
So these are the inverse trig functions for sine, cosine and tangent. What about the other weird ones, cotangent, cosecant and secant? They’re going to follow the same way and I’ll show you what those things look like up next. I’ll see you there.