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4.7 Inverse
Trigonometric
Functions
Inverse Sine Function
Recall that for a function to have an inverse, 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
- domain must be restricted to find its inverse.
y
y = sin x
1


2
x
1
sin x has an inverse
function on this interval
2
The inverse sine function is defined by
y = sin-1x
if and only if
sin y = x.
Angle whose sine is x
The domain of y = sin-1x is [–1, 1].
The range of y = sin-1x is [–/2 , /2].
Example:
1
a. sin 1 
2

6

3
b. arcsin

3
2
 is the angle whose sine is 1 .
6
2
sin   3
3
2
This is another way to write sin-1x .
3
Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse.
y
1

y = cos x

2
x
1
cos x has an inverse
function on this interval
4
The inverse cosine function is defined by
y = cos-1x
if and only if
cos y = x.
Angle whose cosine is x
The domain of y = cos-1x is [–1, 1].
The range of y = cos-1x is [0 , ].
Example:
1
a.) cos 1 
2

b.) arccos  


3
 is the angle whose cosine is 1 .
3
3   5
2  6
2
cos 5   3
6
2
This is another way to write cos-1x.
5
Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse.
y
y = tan x

2
 3
2
3
2
x

2
tan x has an inverse
function on this interval
6
The inverse tangent function is defined by
y = tan-1x
if and only if
tan y = x.
Angle whose tangent is x
The domain of y = tan-1x is (, ) .
The range of y = tan-1x is (–/2 , /2).
Example:
a.) tan
1
3 
6
3
b.) arctan 3 

3
 is the angle whose tangent is
6
3.
3
tan   3
3
This is another way to write tan-1x.
7
Graphing Utility: Graph the following inverse functions.
Set calculator to radian mode.
a. y = sin-1x

–1.5
1.5
–
2
b. y =cos-1x
–1.5
1.5
–

c. y = tan-1x
–3
3
–
8
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. sin-12.5
9
Composition of Functions:
f(f –1(x)) = x
and f –1(f(x)) = x.
Inverse Properties:
If –1  x  1 and – /2  y  /2, then
sin(sin-1x) = x
and sin-1(sin y) = y.
If
–1  x  1 and
0  y  ,
then
cos(cos-1x) = x and cos-1(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(tan-1x) = x and tan-1(tan y) = y.
Example: tan(tan-14) =
10
Example:
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
However, it is coterminal with 5  2   
3
3
5
which does lie in the range of the arcsine
3
x

3
function.
 
 
sin 1 sin 5   sin 1 sin      


3 
3 
3
11

Example:

Find the exact value of tan cos1 2 .
3
adj 2
2
Let  = cos
, then cos  
 .
3 y
hyp 3
1
3
32  22  5
θ


x
2
opp
2
tan cos
 tan  
 5
3
adj
2
1
12