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# 5.1: What’s your Inverse?

Difficulty Level: At Grade Created by: CK-12

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

## Problem 1

Press Y=\begin{align*}Y=\end{align*} and graph Y1=sin1x\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 Y=\begin{align*}Y=\end{align*} and graph Y1=cos1x\begin{align*}Y1 = \cos^{-1}x\end{align*}. Press GRAPH.

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

## Problem 3

Press Y=\begin{align*}Y=\end{align*} and graph Y1=tan1x\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 Y=\begin{align*}Y=\end{align*}.

## Problem 4

Prove cos1x=sec1(1x)\begin{align*}\cos^{-1}x=\sec^{-1}\left(\frac{1}{x}\right)\end{align*}. This will be how you graph y=sec1x\begin{align*}y=\sec^{-1}x\end{align*} in the graphing calculator.

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

## Problem 5

Prove sin1x=csc1(1x)\begin{align*}\sin^{-1}x=\csc^{-1}\left(\frac{1}{x}\right)\end{align*}. This will be how you graph y=csc1x\begin{align*}y=\csc^{-1}x\end{align*} in the graphing calculator.

• Graph your results from above in Y=\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 y\begin{align*}y-\end{align*}axis and a shift of π2\begin{align*}\frac{\pi}{2}\end{align*}. As an equation, it would be cot x=tan(xπ2)\begin{align*}\cot \ x=-\tan \left(x-\frac{\pi}{2}\right)\end{align*}. Take the inverse of y=tan(xπ2)\begin{align*}y=-\tan \left(x-\frac{\pi}{2}\right)\end{align*}.

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

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