Calculus: Inverse Secant, Cosecant, and Cotangent

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Elementary Functions and Their Inverses
Inverse Trigonometric Functions
The Inverse Secant, Cosecant, and Cotangent Functions Page [1 of 2]
Let’s take a look at the reciprocal functions of sine, cosine and tangent. So remember that if you’re looking at , that’s actually cosecant. And what does the cosecant function look like? Well, must remind you what that looks like. The cosecant function has this sort of wacky shape. A lot of asymptotes all over, in fact, there’ll be an asymptote wherever the sine is zero, because remember that cosecant is just . In fact, let me write that down for you, in case you don’t believe me. You're saying, “Wait a minute is that really right?” Absolutely right! csc x = . It’s not the inverse function, it’s one over, it’s the reciprocal function. So whenever this is zero, this thing is going to blow up and we’ll each have an asymptote. You know how the sine function looks. It actually crosses the x-axis at every multiple of ?, and so we have these asymptotes at ?, at 2? and so forth. In fact, if you flip the picture of sine, you actually get this cosecant.
Now, what about the inverse function for cosecant? Well again, we play the same game. We’ve got to find a restricted domain, where the thing is one-to-one, where the thing is either always climbing or always falling. You can certainly see, in general, we have problems, because the curve crosses a vertical line at many places. That’s taboo, it fails the vertical line – I’m sorry, the horizontal line test. The vertical line always is okay, because it’s a function, but the horizontal line test sees that this function is one-to-one. It’s not, so you have to restrict. Well, there are a lot of ways of restricting it. One way to restrict it, for example, would be right in here. Notice that if you increase it any more, it starts to dip down. That’s no good. If I decrease it here, it starts to go up and that’s no good. So there’s one place we could do it, but again, you had to be at the convention. And at the convention they decided we’ll just look from to . And that’s the largest little region where we can actually take an inverse. And what would happen there? Well, if I reflect over this line, you can start to visualize what would happen. This piece here would sort of come down to this area and that little truncated thing would stick up there. And so the picture would look something like this. In fact, it would look exactly like this. This is the arccsc x. So if you flip this, you see this picture.
Now, there’s a couple of interesting things to notice. The first interesting thing to notice is that the y’s only go from up to . And notice the y value y = 0 is actually never attained, because that’s this vertical asymptote that, when flipped, becomes a horizontal asymptote. So, in fact, we have action that goes like this and approaches the x-axis, but never touches, and then below we have the action going up. So, in fact, y will never equal zero here. Also notice that the world ends here. In fact, that’s the end of the world. I don’t want you to think, as with this picture, for example, that the function keeps going here, but here it genuinely stops. You’re seeing the whole picture. And that’s sort of a weird thing, because usually you don’t see functions that just sort of stop in mid-stream. But these are strange functions. These are the inverse functions for these trigonometric things. So this is the graph of arccsc.
What about secant? Well secant, I remind you, is just the reciprocal of cosine. So it has a big graph. So you’ll notice that there’s asymptotes wherever the cosine would be zero, and where is cosine zero? It’s at all the integer multiples of ?, when you have a denominator of 2 there. So , and so forth. So again, we’ve got to find a region where the thing is one-to-one, where the thing can have an inverse. And so, we have a convention where we pick the region between zero and ?. If I go any further, it dips down, if I go any further this way, it’s not there, but I’ll do it live for you. It dips up and so, in fact, this is the biggest region, from zero to ?. And what would that look like, well, just flip along here. So what would happen to this piece? This piece is sort of winging up to here, this piece would wing down to here, and so what we get is this wingy function. This is the function y = arcsec x. And you can see it’s exactly this wing has migrated up to here, this wing here has migrated down to here and there’s this horizontal asymptote. That corresponds with this vertical asymptote, when I switch. Oh, sorry, this is actually – wait, wait, wait. See, I really should have it at the y = x. So this thing wings up to here and this thing wings over to here. And this line corresponds to that line. So that’s the picture. And again, notice how strange this is that the function just sort of stops in mid-air right there. It doesn’t keep going. It’s not like we just didn’t have enough room, we’ve got plenty of room. For this function we’ve got all the room we need in terms of going up and down, because it turns out that the y is only between zero and ?, and again you’ll notice that it’s not defined as . It has that horizontal asymptote there. So if you want to take this function with you, you don’t need a huge suitcase, you just need a suitcase that goes from zero to ?. Now, this way it goes on forever, so it’s got to be a very long suitcase. So you know when you get airline things and they measure the baggage, like the girth around and stuff? So you’re okay this way, but this way you're going to be in trouble, because you're going to start to measure out and you’ll never come back. You’ll just keep going around and around. All right, anyway, there’s the arcsec.
What about the reciprocal of tangent? Well, that’s cotangent, so there’s the cotangent graph. And so, what’s the convention here? Well, the convention here is to look again between and . If I go any further, notice I start to repeat. So this is the largest region where, in fact, we have a one-to-one function. This can be inverted. And then if I put the y = x line down, that will give us a hint as to how this graph would look. And here’s the graph in person. So here is zero. That point zero gets up to here, so it touches the y-axis here, comes down, asymptotic, just like it was asymptotic to the y-axis. When you flip it, it becomes asymptotic to the x-axis, because remember x and y are flipped. So that makes sense. And then, in fact, we have the condition that y only lives between and . And again, we’re missing y = 0, because it’s asymptotic. It has this horizontal asymptote at the x-axis. And so this is the graph of y = arccot x.
Well, so those are the functions. Those are the trig functions, and they’re respective inverses. The thing to remember here is that a trig function just naked, in fact, is not invertible. So you’ve got to close the shades a little bit, they’re modest. Once you close the shades, you can take an inverse. I’ll see you at the next lecture.