Section 4.7
Inverse Trigonometric Functions
A brief review…..
1. If a function is one-to-one, the function has
an inverse.
2. If the graph of a function passes the
horizontal line test, then the function is oneto-one.
3. Some functions can be made to pass the
horizontal line test by restricting their
domains.
More…
4. If (a,b) is a point on the graph of f, then (b,a)
is a point on the graph of f-inverse.
5. The domain of f-inverse is the range of f.
6. The range of f-inverse is the domain of f.
7. The graph of f-inverse is a reflection of the
graph of f about the line y = x.
The inverse sine function
• Denoted by sin-1
• The domain of y = sin x is restricted to


2
x

2
• y = sin-1 x means that sin y = x.
• sin-1 x is the angle, between –π/2 and π/2,
inclusive, whose sine value is x.
The inverse cosine function
• Denoted by cos-1
• The domain of y = cos x is restricted to
0 x 
• y = cos-1 x means that cos y = x.
• cos-1 x is the angle, between 0 and π, inclusive,
whose cosine value is x.
The inverse tangent function
• Denoted by tan-1
• The domain of y = tan x is restricted to


2
x

2
• y = tan-1 x means that tan y = x.
• tan-1x is the angle, between –π/2 and π/2,
whose tangent value is x.
Evaluating inverse functions
• For exact values, use your table and/or your
knowledge of the unit circle.
• For approximate values, use your calculator
(be careful to watch your MODE).
Examples
sin
1
1
2
cos
1
 0.46
1
cos 1
sin
tan
1
 1
1
6
5
Evaluating composite functions
• Composite functions come in two types:
1. The function is on the “inside”.
2. The inverse is on the “inside”.
• In either case, work from the “inside out”.
• Be sure to observe the restricted domains of the
functions you are dealing with.
• Sometimes the function and inverse will “cancel” each
other but, again, watch your restricted domains.
• For values not on the unit circle, draw a sketch and
use right triangle trigonometry.
Examples
 1  
sin  sin

4


tan tan 1 6
 1 7 
tan  cos

25 


3 

tan  tan

4 

1
 1  1 
sin cos  
 2 

 1  5 
sec sin   
 8 

 1  5 
cos  tan   
 7 

Weird Examples
• Use a right triangle to write the following
expression as an algebraic expression:

1
cos sin 7 x

 1 2 
cos sin

x
