# 9 Inverse Trigonometric Functions

### Learning Objectives

In this section, you will:

- Understand and use the inverse sine, cosine, and tangent functions.
- Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
- Use a calculator to evaluate inverse trigonometric functions.
- Find exact values of composite functions with inverse trigonometric functions.

Recall the definition of an inverse function:

Let be a function. The inverse of f, denoted by is the set of ordered pairs {(𝑏,𝑎)∈𝐵×𝐴 | 𝑓(𝑎)=𝑏}. In essence, the inverse function reverses the action of the function.

For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.

### Understanding and Using the Inverse Sine, Cosine, and Tangent Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in (Figure).

For example, if then we would write Be aware that does not mean The following examples illustrate the inverse trigonometric functions:

- Since then
- Since then
- Since then

In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if then an inverse function would satisfy

Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. (Figure) shows the graph of the sine function limited to and the graph of the cosine function limited to

(Figure) shows the graph of the tangent function limited to

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.

On these restricted domains, we can define the inverse trigonometric functions.

- The inverse sine function means The inverse sine function is sometimes called the arcsine function, and notated
- The inverse cosine function means The inverse cosine function is sometimes called the arccosine function, and notated
- The inverse tangent function means The inverse tangent function is sometimes called the arctangent function, and notated

The graphs of the inverse functions are shown in (Figure), (Figure), and (Figure). Notice that the output of each of these inverse functions is a *number, *an angle in radian measure. We see that has domain and range has domain and range and has domain of all real numbers and range To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line

### Relations for Inverse Sine, Cosine, and Tangent Functions

For angles in the interval if then

For angles in the interval if then

For angles in the interval if then

### Writing a Relation for an Inverse Function

Given write a relation involving the inverse sine.

## Show Solution

Use the relation for the inverse sine. If then .

In this problem, and

### Try It

Given write a relation involving the inverse cosine.