In order for inverses of functions to be functions, the original function must pass the **horizontal line test**. Though none of the trigonometric functions pass the horizontal line test, you can restrict their domains so that they can pass. Then the inverses are produced just like with normal functions. Once you have the basic inverse functions, the normal transformation rules apply.

Why is

### Graphs of Inverse Trigonometric Functions

Since none of the six trigonometric functions pass the horizontal line test, you must **restrict their domains** before finding inverses of these functions. This is just like the way

Consider the sine graph:

As a general rule, the restrictions to the domain are either the interval

The result of this inversion is that arcsine will only ever produce angles between

The blue curve below shows

The portion of cosine that fits the horizontal line test is the interval

### Examples

#### Example 1

Earlier, you were asked why sine and arcsine don't always just cancel out. Since arcsine only produces angles between

#### Example 2

Graph the function

Since the graph of

#### Example 3

Evaluate the following expression with and without a calculator using right triangles and your knowledge of inverse trigonometric functions.

When using a calculator it can be extremely confusing trying to tell the difference between

The hardest part of this question is seeing the csc as a function (which produces an angle) on a ratio of a hypotenuse of 13 and an opposite side of -5. The sine of the inverse ratio must produce the same angle, so you can substitute it.

csc−1(−135)=sin−1(−513) cot(θ)=1tanθ

Not using a calculator is usually significantly easier. Start with your knowledge that

Cotangent is adjacent over opposite.

#### Example 4

What is the graph of

Graph the portion of tangent that fits the horizontal line test and reflect across the line

#### Example 5

What is the graph of

Graph the portion of cosecant that fits the horizontal line test and reflect across the line

Note that

### Review

1. Graph

2. Graph

Name each of the following graphs.

3.

4.

5.

6.

7.

Graph each of the following functions using your knowledge of function transformations.

8.

9.

10.

11.

12.

13.

14.

15. \begin{align*}t(x)=\csc^{-1}(x+1) -\frac{3 \pi}{2}\end{align*}

16. \begin{align*}v(x)=2 \sec^{-1}(x+2)+\frac{\pi}{2}\end{align*}

17. \begin{align*}w(x)=-\cot^{-1}(x)-\frac{\pi}{2}\end{align*}

Evaluate each expression.

18. \begin{align*}\sec \left(\tan^{-1} \left[\frac{3}{4}\right]\right)\end{align*}

19. \begin{align*}\cot \left(\csc^{-1} \left[\frac{13}{12}\right]\right)\end{align*}

20. \begin{align*}\csc \left(\tan^{-1} \left[\frac{4}{3}\right]\right)\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.8.