If you restrict to the interval , the function *increases*:

This implies that the function is *one-to-one*, and hence it
has an inverse. The inverse is called the * inverse
sine* or * arcsine function*, and is denoted
or . Note that in the second case
*does not mean* " "!

Thus, is *the angle whose sine is x*.
Another way of saying this is:

The fact that and are inverse functions can be expressed by the following equations:

Since the restricted takes angles in the range and produces numbers in the range , takes numbers in the range and produces angles in the range .

* Example.*

Sine and arcsine are inverses, so they undo one another --- but you have to be careful!

can't be , because always returns an angle in the range .

* Example.* Find .

First, let . This means that . Now , so I get the following picture:

I got the adjacent side using Pythagoras: .

Using the triangle, I have

You can find a derivative formula for using implicit differentiation. Let . This is equivalent to . Differentiate implicitly:

I'd like to express the result in terms of x. Here's the right triangle that says :

I found the other leg using Pythagoras. You can see that . Hence, . That is,

Every derivative formula gives rise to a corresponding antiderivative formula:

Before I do some calculus examples, I want to mention some of the
other inverse trig functions. I'll discuss the *
inverse cosine*, * inverse tangent*, and * inverse secant* functions.

- You get the inverse cosine by inverting , restricted to .

- You get the inverse tangent by inverting , restricted to .

- You get the inverse secant by inverting , restricted to together with .

As with and , the domains and ranges of these functions and their inverses are "swapped":

* Example.*

You can derive the derivative formulas for the other inverse trig functions using implicit differentiation, just as I did for the inverse sine function.

* Example.* Derive the formula for .

The derivation starts out like the derivation for . Let , so . Differentiating implicitly, I get

There are two cases, depending on whether or .

Suppose . Then is in the interval , as illustrated in the first diagram above. You can see from the picture that

Therefore,

, so x is positive, and . Therefore,

Now suppose that . Then is in the interval , as illustrated in the second diagram above. Since x is negative, the hypotenuse must be , since it must be positive and since must equal x. In this case,

Therefore,

, so x is negative, and . Therefore,

This proves that in all cases.

* Example.*

I don't need absolute values in the last example, because is always positive.

* Example.*

Hence,

A function with zero derivative is constant, so

But when ,

So I get the identity

Here are the integration formulas for some of the inverse trig functions. I'm giving extended versions of the formulas --- with " " replacing the "1" that you'd get if you just reversed the derivative formulas --- in order to save you a little time in doing problems.

* Example.* Derive the extended integral formula from the formula .

* Example.* Using the formula with
,

Using the formula with ,

* Example.*

* Example.*

* Example.*

* Example.*

Copyright 2005 by Bruce Ikenaga