Download CLASS 1 CHARACTERISTICS of FUNCTIONS, ALGEBRAICALLY

Chapter 5 – Trigonometric Functions:
Unit Circle Approach
5.5 - Inverse Trigonometric Functions & Their
Graphs
Review of Inverse Functions
Remember
 If the graph passes the horizontal line test, then the
function has an inverse functions.

If a point (a, b) is on the graph of f, then the point
(b, a) is on the graph of f -1.

The graph of f -1 is a reflection of the graph of f about
the line y=x.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Sine Function


Does not pass the horizontal line test.
Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Definition

The inverse sine function is the function sin-1 with
domain [-1, 1] and range [- ⁄ 2,  ⁄ 2] defined by
1
sin x  y  sin y  x

The inverse sine function is also called arcsine
denoted by arcsin.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Note
5.5 - Inverse Trigonometric Functions & Their
Graphs
Graph of Inverse sine
5.5 - Inverse Trigonometric Functions & Their
Graphs
Cancellation Properties - Sine

Thus y = sin-1x is the number in the interval
[- ⁄ 2,  ⁄ 2] whose sine is x.

In other words we have the following:


sin sin 1 x  x
sin
1
 sin x   x
for
for
1  x  1


2
5.5 - Inverse Trigonometric Functions & Their
Graphs
x

2
Examples
Find the exact value of the following:
2
1. arcsin
2
2. sin
1
3
2
 1
3. arcsin   
 2
1
4. sin 2
5.5 - Inverse Trigonometric Functions & Their
Graphs
Cosine Function


Does not pass the horizontal line test.
Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Definition

The inverse cosine function is the function cos-1
with domain [-1, 1] and range [0, ] defined by
1
cos x  y  cos y  x

The inverse sine function is also called arccosine
denoted by arccos.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Graph of Inverse Cosine
5.5 - Inverse Trigonometric Functions & Their
Graphs
Cancellation Properties - Cosine

Thus y = cos-1x is the number in the interval [0, ]
whose cosine is x.

In other words we have the following:


cos cos1 x  x
for
1  x  1
cos1  cos x   x
for
0 x 
5.5 - Inverse Trigonometric Functions & Their
Graphs
Examples
Find the exact value of the following:
2
1. arccos
2
2. cos
1
3
2
 1
3. arccos   
 2

3
4. cos  

 2 
1
5.5 - Inverse Trigonometric Functions & Their
Graphs
Tangent Function


Does not pass the horizontal line test.
Must restrict the domain to create an inverse
function.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Definition

The inverse tangent function is the function tan-1
with domain (-∞, ∞) and range (- ⁄ 2,  ⁄ 2) defined
by
1
tan x  y  tan y  x

The inverse tangent function is also called
arctangent denoted by arctan.
5.5 - Inverse Trigonometric Functions & Their
Graphs
Graph of Inverse Tangent
5.5 - Inverse Trigonometric Functions & Their
Graphs
Cancellation Properties Tangent

Thus y = tan-1x is the number in the interval
(- ⁄ 2,  ⁄ 2) whose sine is x.

In other words we have the following:


tan tan 1 x  x
tan
1
 tan x   x
for
for
  x  


2
5.5 - Inverse Trigonometric Functions & Their
Graphs
x

2
Examples
Find the exact value of the following:
1. arctan 3
2. tan 1  1
5.5 - Inverse Trigonometric Functions & Their
Graphs
Evaluating Compositions
5.5 - Inverse Trigonometric Functions & Their
Graphs
Inverse Properties
5.5 - Inverse Trigonometric Functions & Their
Graphs
Using Inverse Properties
Evaluate the following:
 
1. sin  sin 
4

1
7 
1 
2. sin  sin

4 

3. tan  tan
1
 5  
4. cos  cos  
5.5 - Inverse Trigonometric Functions & Their
Graphs
1
Examples – pg. 412

Find the exact value of the expression if it is defined.
 1 1 
39. tan  sin

2


40. cos sin 0
 1 3 
41. cos  sin

2 

 1 2 
42. tan  sin

2 


5.5 - Inverse Trigonometric Functions & Their
Graphs
1
