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 .
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.
As with and , the domains and ranges of these functions and their inverses are "swapped":
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
, 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,
, so x is negative, and . Therefore,
This proves that in all cases.
I don't need absolute values in the last example, because is always positive.
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 ,
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