Topic: Graphs of Trigonometric Functions

Subtopic: Inverse Trigonometric Functions

**Overview**

We have worked with inverse trig functions (such as tan^{-1}(1/√3)=π/6) before, but primarily just evaluating them on a calculator or via the unit circle. In this lesson we formally define inverse trig functions and extend our knowledge about them including their exploring their graphs, domains, and ranges.

Before studying inverse trigonometric functions, I strongly recommend that you review inverse functions in general. See the supplementary resources below for review sites.

It is very important to remember that while regular functions take an angle input and output a ratio, inverse trig functions take a ratio input and output the corresponding angle. The answer from an inverse trig function is an angle. Note the units (degrees or radians) in which the output angle needs to be.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 4.7.1 Graph the three main inverse trig functions arcsin(x), arccos(x), arctan(x)
- 4.7.2 Know the domain and range of all six inverse trig functions
- 4.7.3 Know in which quadrants the answers from each of the six inverse trig functions lie (important!)
- 4.7.4 Evaluate inverse trig functions manually
- 4.7.5 Evaluate compositions of trig and inverse trig functions both numeric, such as tan(cos
^{-1}(-√3/2)), and algebraic, such as sec(sin^{-1}(x))

**Terminology**

Define: vertical and horizontal line tests, one-to-one trig function, inverse trigonometric function

**Supplementary Resources (optional)**

Need a general review of inverse functions? Patrick Just Math Tutorial's Inverse Functions - The Basics is a comprehensive (rather long) video with a good variety of algebraica processes related to inverse functions.

And from James Sousa's MathIsPower4U come lots of videos and examples, for you!

Mini-Lesson on general inverse functions:

__Inverse Functions__

__Animation: Illustrate why a function must be one-to-one to have an inverse function__

Mini-Lessons on trigonometric inverse functions:

__Intro to the Inverse Functions of Sine, Cosine, and Tangent__

__Evaluating Expressions and Solving Problems Using Inverse Sine, Cosine, and Tangent__

__Animation: Illustrate why the domain must be restricted__

__Intro to the Inverse Functions of Cosecant, Secant, and Cotangent__

__Evaluating Expressions Involving Inverse Cosecant, Secant, and Cotangent__

Examples:

__Examples: Evaluate Inverse Trig Expressions (Part 1)__

__Examples: Evaluate Inverse Trig Expressions (Part 2)__

__Examples: Evaluate Inverse Trig Expressions (Part 3)__

__Examples: Evaluate Expression Involving Inverse Trig Functions (Part 1)__

__Examples: Evaluate Expression Involving Inverse Trig Functions (Part 2)__

__Ex 0: Evaluate __arccot__(-3.6) Using a Calculator__

__Ex 1: Evaluate tan(__arcsin__(-12/13))__

__Ex 2: Evaluate sin(__arctan__(-7))__

__Ex 3: Evaluate sin(__arctan__(u/3))__