Topic: Graphs of Trigonometric Functions
Subtopic: Inverse Trigonometric Functions
We have worked with inverse trig functions (such as `tan^-1(1/sqrt3)=pi/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.
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(-sqrt3/2))`, and algebraic, such as `sec(sin^-1(x))`
Terms you should be able to define: vertical and horizontal line tests, one-to-one trig function, inverse trigonometric function
Mini-Lectures and Examples
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 algebraic 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:
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: 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))