## Skills Review for Integrals Resulting in Inverse Trigonometric Functions

### Learning Outcomes

• Evaluate inverse trigonometric functions

In the Integrals Resulting in Inverse Trigonometric Functions section, we will need to know how to evaluate inverse trigonometric functions. This concept is reviewed here.

## Evaluate Inverse Trigonometric 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 below.

For example, if $f(x)=\sin x$, then we would write $f^{-1}(x)={\sin}^{-1}{x}$. Be aware that ${\sin}^{-1}x$ does not mean $\frac{1}{\sin{x}}$. The following examples illustrate the inverse trigonometric functions:

• Since $\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$, then $\frac{\pi}{6}=\sin^{−1}(\frac{1}{2})$.
• Since $\cos(\pi)=−1$, then $\pi=\cos^{−1}(−1)$.
• Since $\tan\left(\frac{\pi}{4}\right)=1$, then $\frac{\pi}{4}=\tan^{−1}(1)$.

### A General Note: Relations for Inverse Sine, Cosine, and Tangent Functions

For angles in the interval $\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right]$, if $\sin y=x$, then $\sin^{−1}x=y$.

For angles in the interval [0, π], if $\cos y=x$, then $\cos^{−1}x=y$.

For angles in the interval $\left(−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right)$, if $\tan y=x$, then $\tan^{−1}x=y$.

Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically $\frac{\pi}{ 6} (30^\circ)\text{, }\frac{\pi}{ 4} (45^\circ),\text{ and } \frac{\pi}{ 3} (60^\circ)$, and their reflections into other quadrants.

### How To: Given a “special” input value, evaluate an inverse trigonometric function.

1. Find angle x for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
2. If x is not in the defined range of the inverse, find another angle y that is in the defined range and has the same sine, cosine, or tangent as x, depending on which corresponds to the given inverse function.

### Example: Evaluating Inverse Trigonometric Functions for Special Input Values

Evaluate each of the following.

a. $\sin−1\left(\frac{1}{2}\right)$

b. $\sin−1\left(−\frac{2}{\sqrt{2}}\right)$

c. $\cos−1\left(−\frac{3}{\sqrt{2}}\right)$

d. $\tan^{− 1}(1)$

### Try It

Evaluate each of the following.

1. $\sin^{−1}(−1)$
2. $\tan^{−1}(−1)$
3. $\cos^{−1}(−1)$
4. $\cos^{−1}(\frac{1}{2})$