# 5.1: What’s your Inverse?

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

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

**Time Required: 30 minutes**

## Activity Overview

This activity is intended to be teacher-led, where students will use their graphing calculators to graph the six inverse trig functions. The inverse reciprocal properties will be derived as a class.

**Topics Covered**

- Graphing inverses
- Finding the inverse of a function

**Teacher Preparation and Notes**

- Make sure students have cleared \begin{align*}Y=\end{align*} menu before starting.
- Students will need to know how to find the inverse of a function algebraically. It might be helpful to have a warm-up covering this topic as a quick review.
- Go over the definition of a restricted domain and make sure students understand why they are necessary to find the inverse of trigonometric functions.
- Make sure students’ calculators are in
**Radians**.

**Associated Materials**

- Student Worksheet: What's your Inverse http://www.ck12.org/flexr/chapter/9702

## Problem 1

For the domain and range of \begin{align*}y = \sin^{-1}x\end{align*}, review what the domain and range are of \begin{align*}y = \sin x\end{align*}. The domain is all real numbers and the range is between -1 and 1. For the inverse, they will be switched, so the domain is between -1 and 1 and the range should be all real numbers. However, from looking at the graph, we know this is not true. Hence, there is a restricted domain on \begin{align*}y = \sin x\end{align*} so that it can have an inverse (recall that \begin{align*}y = \sin x\end{align*} is periodic). The domain is restricted to \begin{align*}[-\frac{\pi}{2}, \frac{\pi}{2}]\end{align*}, and the range of \begin{align*}y = \sin^{-1}x\end{align*} would be between \begin{align*}-\frac{\pi}{2}\end{align*} and \begin{align*}\frac{\pi}{2}\end{align*}.

## Problem 2

Again, review the domain and range of \begin{align*}y = \cos x\end{align*}, which are all real numbers for the domain and between -1 and 1 for the range. By looking at the graph, domain of the inverse is also between -1 and 1 and the range is between 0 and \begin{align*}\pi\end{align*}.

## Problem 3

The domain of \begin{align*}y = \tan x\end{align*} is all real numbers, except for every odd multiple of \begin{align*}\frac{\pi}{2}\end{align*} \begin{align*}\left(x \ne \frac{\pi}{2},\frac{3 \pi}{2}, \frac{5 \pi}{2},\ldots \right)\end{align*}. The range is all real numbers. So, the range of \begin{align*}y = \tan^{-1}x\end{align*} is between two of these asymptotes \begin{align*}\left (-\frac{\pi}{2}, \frac{\pi}{2} \right )\end{align*} and there are horizontal asymptotes. The domain is all real numbers.

**For secant, cosecant and cotangent guide students through how to derive the equation that is needed to plug into the calculator.**

## Problem 4

Prove \begin{align*}\cos^{-1}x = \sec^{-1} \left(\frac{1}{x} \right)\end{align*} . *Walk students through these steps*.

\begin{align*}y &= \cos^{-1}x\\ \cos y &= x\\ \frac{1}{\cos y} &= \frac{1}{x}\\ \sec y &= \frac{1}{x}\\ y &= \sec^{-1} \left(\frac{1}{x} \right)\end{align*}

This means that \begin{align*}\cos^{-1}x = \sec^{-1}\left(\frac{1}{x}\right) \end{align*} and \begin{align*}\sec^{-1}x = \cos^{-1}\left(\frac{1}{x}\right)\end{align*}.

In \begin{align*}Y1\end{align*}, students should input \begin{align*}\cos^{-1}\left(\frac{1}{x} \right)\end{align*} in order to graph \begin{align*}y = \sec^{-1}x\end{align*}.

## Problem 5

Prove \begin{align*}\sin^{-1}x = \csc^{-1}\left(\frac{1}{x}\right)\end{align*}. *Walk students through these steps*.

\begin{align*}y &= \sin^{-1}x\\ \sin y &= x\\ \frac{1}{\sin y} &= \frac{1}{x}\\ \csc y &= \frac{1}{x}\\ y &= \csc^{-1} \left(\frac{1}{x}\right)\end{align*}

This means that \begin{align*}\sin^{-1}x = \csc^{-1}\left(\frac{1}{x} \right)\end{align*} and \begin{align*}\csc^{-1}x = \sin^{-1}\left(\frac{1}{x} \right)\end{align*}.

In \begin{align*}Y1\end{align*}, students should input \begin{align*}\sin^{-1}\left(\frac{1}{x} \right)\end{align*} in order to graph \begin{align*}y = \csc^{-1}x\end{align*}.

## 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, the relationship would be \begin{align*}\cot x = - \tan \left(x -\frac{\pi}{2}\right )\end{align*}. Students will need to take the inverse of \begin{align*}y = - \tan \left(x - \frac{\pi}{2}\right )\end{align*} to find how to graph \begin{align*}y = \cot^{-1}x\end{align*} in their calculators.

\begin{align*}y &= - \tan \left(x - \frac{\pi}{2}\right)\\ x &= - \tan \left(y - \frac{\pi}{2}\right)\\ -x &= \tan \left(y - \frac{\pi}{2}\right)\\ \tan^{-1}(-x) &= y - \frac{\pi}{2}\\ \frac{\pi}{2} + \tan^{-1}(-x) &= y\\ \frac{\pi}{2} - \tan^{-1}x &= y\end{align*}

This means that \begin{align*}\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x\end{align*} and \begin{align*}\tan^{-1} x = \frac{\pi}{2} - \cot^{-1}x\end{align*}.

Because tangent is an odd function, or \begin{align*}\tan (-x) = - \tan x\end{align*}, then its inverse is also odd. In \begin{align*}Y1\end{align*} students should input \begin{align*}\frac{\pi}{2} - \tan^{-1}x\end{align*} in order to graph \begin{align*}y = \cot^{-1} x\end{align*}