Inverse Trigonometric
Functions
Inverse Trig Functions Objectives
• Evaluate inverse sine, cosine, and tangent
functions.
• Evaluate compositions of inverse trig functions.
• Use inverse trig functions in applications.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
Definitions and Terminology
-1
sin x or arcsinx is the angle whose sine is x.
  sin x  x  sin 
1
-1
cos x or arccosx is the angle whose cosine is x.
  cos x  x  cos 
1
-1
tan x or arctanx is the angle whose tan is x.
  tan x  x  tan 
1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
Restricting the Sine Function
y
y = sin x


2
x
1
Sin x has an inverse
function on this interval.
Recall that for a function to have an inverse on its entire
domain, it must be a one-to-one function and pass the
Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test and must
be restricted to for its inverse to be a function.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
7
The inverse sine function is defined by
y = arcsin x
if and only if
sin y = x.
(0,1)
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–/2 , /2].
(0,–1)
Examples:
a. arcsin 1  
6
2
b. sin 1 3 
2

3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
x
 is the angle whose sine is 1 .
6
2
sin    3
3
2
8
Restricting the Cosine Function
pair/share
f(x) = cos x must be restricted to find its inverse.
y
y = cos x


2
x
1
Cos x has an inverse
function on this interval.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9
The inverse cosine function is defined by
y = arccos x if and only if
cos y = x.
Angle whose cosine is x
y
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , ].
(0,-1)
(0,1)
Examples:
a.) arccos 1  
2 3
 is the angle whose cosine is 1 .

 5
3
b.) cos  


6
 2 
cos 5   3
6
2
1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
3
2
10
The inverse tangent function is defined by
y = arctan x
if and only if
tan y = x.
(0,1)
Angle whose tangent is x
The domain of y = arctan x is (, ) .
x
The range of y = arctan x is [–/2 , /2].
(0,–1)
Example:
a.) arctan 3  
3
6


b.) tan  1  4
1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
 is the angle whose tangent is
6
3.
3
 
tan    1
4
12
Graphing Utility: Graph the following inverse functions.
Set calculator to radian mode.
a. y = arcsin x

–1.5
1.5
–
2
b. y = arccos x
–1.5
1.5
–

c. y = arctan x
–3
3
–
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
13
Graphing Utility: Approximate the value of each expression.
Set calculator to radian mode.
a. cos–1 0.75
b. arcsin 0.19
c. arctan 1.32
d. arcsin 2.5
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
15
Inverse Properties:
If –1  x  1, then sin(arcsin x) = x
and if – /2  y  /2, then arcsin(sin y) = y.
If –1  x  1, then cos(arccos x) = x
and if 0  y  , then arccos(cos y) = y.
If x is a real number, then tan(arctan x) = x
and if –/2 < y < /2, then arctan(tan y) = y.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
17
Examples:
tan  arctan 4  4
4

cos  cos
3

1
 2

3

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
18
You try:
a. sin–1(sin (–/2)) = –/2
 
b. sin 1 sin 5 

3 
5 does not lie in the range of the arcsine function, –/2  y  /2.
3
y
5
3
x

3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
 
sin 1 sin 5    

3 
3
19
Evaluating Composition of Functions


Find the exact value of tan arccos 2 .
3
y
3
5
u


x
2
opp
2
tan arccos  tan u 
 5
3
adj
2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
20
You Try:

 3 
Find the exact value of: cos  arcsin    
 5 

y

Precalculus
x
21
4.7 Inverse Trigonometric
Write each of the following as an algebraic expression in x:
a) sin  arccos 3x  0  x  1
3
b) cot  arccos 3x  0  x  1
3
y
x
a) 1  9x 2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
b)
3x
1  9x 2
22
Applying Inverse Trig Functions:
A person walks 120 ft. away from a building. The line
of sight to the top of the building is 150 ft. What is the
angle of elevation to the top of the building?
  36.87

120 ft
H = 74.98 ft
A person stands 50 ft. from a tree. If the height of the tree is
70 ft., find the angle of elevation to the top of the tree.
  54.46

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
24
Homework
4.7 p 316:
1-7 odd, 13-53 odd
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
26