Chapter 4
Trigonometric Functions
1
4.7 Inverse Trigonometric
Functions
Objectives:
Evaluate inverse sine functions.
Evaluate other inverse trigonometric
functions.
Evaluate compositions of trigonometric
functions.
2
Inverse Functions
Recall that a function and its inverse
reflect over the line y = x.
What must be true for a function to have
an inverse?
 It must be one-to-one, that is, it must
pass the horizontal line test.
3
More Inverse Functions
Are sine, cosine, and tangent one-to-one?
If not, what must we do so that these
functions will have inverse functions?
Hint: Consider y = x2.
We must restrict the domain of the
original function.
4
Sine and Its Inverse
f(x) = sin x does not pass the Horizontal
Line Test
It must be restricted to find its inverse.
y
1

y = sin x

2
x
1
Sin x has an inverse


 x
function on this
2
2
interval.
5
Inverse Sine Function
The inverse sine function is defined by
y = arcsin x if and only if
sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is _____________.
 Why are the domain and range defined
this way?
6
What Does “arcsin” Mean?
In an inverse function, the x-values and the
y-values are switched.
So, arcsin x means the angle (or arc) whose
sin is x.
Notation for inverse sine
 arcsin x
 sin -1 x
7
Examples
If possible, find the exact value.
 1
1. arcsin   
 2
2. sin
1
3
2
3. sin 1 2
8
Graphing Arcsine
Create a table for sin y = x for –π/2 ≤ y ≤ π/2.
y
x
–π/2
–π/4
–π/6
0
π/6
π/4
π/2
Graph x on horizontal axis and y on vertical
axis.
9
Graph of Arcsine
10
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.
0 x 
11
Inverse Cosine Function
The inverse cosine function is defined by
y = arccos x if and only if cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0, π].
Notation for inverse cosine:

arccos x or cos -1 x
12
Examples
If possible, find the exact value
1
1. arccos 
2

3

2. cos  

2


1
13
Graphing Arccos
Create a table for cos y = x for 0 ≤ y ≤ π.
y
x
0
π/6
π/3
π/2
2π/3
5π/6
π
Graph x on horizontal axis and y on vertical
axis.
14
Graph of Arccos
15
Inverse Tangent Function
f(x) = tan x must be restricted to find its
y
inverse.
y = tan x

2
 3
2
3
2 x

2
Tan x has an inverse
function on this interval.


2
x

2
16
Inverse Tangent Function
The inverse tangent function is defined by
y = arctan x if and only if
tan y = x.
Angle whose tangent is x
The domain of y = arctan x is (–∞, ∞).
The range of y = arctan x is (–π/2, π/2).
Notation for inverse tangent:

arctan x or tan -1 x
17
Examples
If possible, find the exact value
 3

1. arctan 

3


 
2. tan 1 3
18
Graph of Arctan
19
Examples
Evaluate using your calculator. (What mode
should the calculator be in?)
1. cos 1 0.75
2. arcsin 0.19
3. arctan 1.32
4. arcsin 2.5
20
Summary
21
Composition of Functions
Given the restrictions specified in the previous
slide, we have the following properties of
inverse trig functions.
sin(arcsin x)  x and arcsin(sin y )  y
cos(arccos x)  x and arccos(cos y )  y
tan(arctan x)  x and arctan(tan y )  y
22
Examples
If possible, find the exact value.
1. tan arctan  5
 5 
2. arcsin  sin

3 


3. cos cos 1 

23
Example


Find the exact value of tan arccos 2 .
3
adj 2
2
Let u = arccos , then cos u 
 .
3
hyp 3
y
3
32  22  5
u

x

2
opp
2
tan arccos  tan u 
 5
3
adj
2
24
Example

 3 
Find the exact value of cos arcsin    .
 5 

25
Homework 4.7
Worksheet 4.7
26