# 5.1: What’s your Inverse?

**At Grade**Created by: CK-12

*This activity is intended to supplement Trigonometry, Chapter 4, Lesson 3.*

## Problem 1

Press \begin{align*}Y=\end{align*} and graph \begin{align*}Y1 = \sin^{-1} x\end{align*}. Press **MODE** and make sure **Radian** is highlighted. Press **GRAPH**.

- Press
**ZOOM**,**7:ZTrig**. Graph and determine the domain and range of the function.

- Why is there a restricted domain on this function?

## Problem 2

Press \begin{align*}Y=\end{align*} and graph \begin{align*}Y1 = \cos^{-1}x\end{align*}. Press **GRAPH**.

- Graph and determine the domain and range of the function.

## Problem 3

Press \begin{align*}Y=\end{align*} and graph \begin{align*}Y1 = \tan^{-1}x\end{align*}. Press *'GRAPH*.

- Graph and determine the domain and range of the function.

For secant, cosecant and cotangent, it is a little more difficult to plug into \begin{align*}Y=\end{align*}.

## Problem 4

Prove \begin{align*}\cos^{-1}x=\sec^{-1}\left(\frac{1}{x}\right)\end{align*}. This will be how you graph \begin{align*}y=\sec^{-1}x\end{align*} in the graphing calculator.

- Graph your results from above in \begin{align*}Y=\end{align*}. Find the domain and range of the function.

## Problem 5

Prove \begin{align*}\sin^{-1}x=\csc^{-1}\left(\frac{1}{x}\right)\end{align*}. This will be how you graph \begin{align*}y=\csc^{-1}x\end{align*} in the graphing calculator.

- Graph your results from above in \begin{align*}Y=\end{align*}. Find the domain and range of the function.

## Problem 6

Tangent and cotangent have a slightly different relationship. Recall that the graph of cotangent differs from tangent by a reflection over the \begin{align*}y-\end{align*}axis and a shift of \begin{align*}\frac{\pi}{2}\end{align*}. As an equation, it would be \begin{align*}\cot \ x=-\tan \left(x-\frac{\pi}{2}\right)\end{align*}. Take the inverse of \begin{align*}y=-\tan \left(x-\frac{\pi}{2}\right)\end{align*}.

- Graph your results from above in \begin{align*}Y=\end{align*}. Find the domain and range of the function.

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