L> The Inverse Sine Function Understanding theInverse Sine Function”> The Function y = sin-1x = arcsin x and its Graph: Since y = sin -1x is the inverse of the function y = sin x, the function **y = sin-1x if and only if sin y = x**. But, since y = sin x is not one-to-one, its domain must be restricted in order that y = sin-1x is a function. To get the graph of y = sin-1x, start with a graph of y = sin x. Restrict the domain of the function to a one-to-one region – typically” width=”57″ height=”39″ align=”absmiddle”> is used (highlighted in red at right) for sin-1x. This leaves the range of the restricted function unchanged as <-1, 1>. Reflect this graph across the line y = x to get the graph of y = sin-1x (y = arcsin x), the black curve at right. Notice that y = sin-1x has domain <-1, 1> and range ” width=”56″ height=”39″ align=”absmiddle”>. It is strictly increasing on its entire domain. So, when you ask your calculator to graph y = sin-1x, you get the graph shown at right. (The viewing window is <-2, 2> x <-2, 2>.)

## Evaluating y = sin-1x:

### Example 1: Evaluate sin-1(1/2)

Most people are more familiar (and more comfortable) with the trigonometric functions than their inverses. Therefore, the first step in evaluating this expression is to say that if y = sin-1(1/2), then sin y = 1/2.

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This simple trigonometric function has an infinite number of solutions:

Five of these solutions are indicated by vertical lines on the graph of y = sin x below.

So, is the value of sin-1 (1/2) given by the expressions above? No! It is vitally important to keep in mind that the inverse sine function is a single-valued, one-to-one function. Only one of the infinite number of solutions given above is the result we want. Which one? Remember that the range of sin-1x is ” width=”56″ height=”39″ align=”absmiddle”>, which is indicated in blue in the figure above. **It is really important to know the domain and range of the inverse trigonometric functions!** (Why is this blue interval marked on the x-axis if it represents the range of sin-1x? Because the

*range*of the inverse function equals the

*domain*of the principal function.) The only solution of y = sin x that falls within the required range is

(the solid red line in the figure above). Therefore,

### Example 2: What is

A unit-circle diagram is shown at right. Notice that candidates for the solution include:

However, only one of these values is in the range of sin-1x (” width=”56″ height=”39″ align=”absmiddle”>), so:

The Derivative of y = sin-1x:

The derivative of y = sin-1 x is: (Click here for a derivation.)

The graphs of y = sin-1 x and its derivative is shown at right. The domain of y” is (-1. 1). Since y = sin-1 x is always increasing, y” > 0 for all x in its domain.

## Integrals Involving the Inverse Sine Function

Since

,

. This means that the arcsine function arises in discussions involving integrals (and areas) of “relatively normal looking” algebraic functions. For instance:

This is the shaded area shown in the TI-89 screen shot at right. (The window is <-0.5, 1.1> x <0, 3>.)