# 5.1: What’s your Inverse?

*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 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 , review what the domain and range are of . 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 so that it can have an inverse (recall that is periodic). The domain is restricted to , and the range of would be between and .

## Problem 2

Again, review the domain and range of , 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 .

## Problem 3

The domain of is all real numbers, except for every odd multiple of . The range is all real numbers. So, the range of is between two of these asymptotes 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 . *Walk students through these steps*.

This means that and .

In , students should input in order to graph .

## Problem 5

Prove . *Walk students through these steps*.

This means that and .

In , students should input in order to graph .

## Problem 6

Tangent and cotangent have a slightly different relationship. Recall that the graph of cotangent differs from tangent by a reflection over the axis and a shift of . As an equation, the relationship would be . Students will need to take the inverse of to find how to graph in their calculators.

This means that and .

Because tangent is an odd function, or , then its inverse is also odd. In students should input in order to graph