4.7 INVERSE TRIGONOMETRIC FUNCTIONS
Copyright © Cengage Learning. All rights reserved.
2
• Evaluate and graph the inverse sine function.
• Evaluate and graph the other inverse
trigonometric functions.
• Evaluate and graph the compositions of
trigonometric functions.
What You Should Learn
3
Inverse Sine Function
4
Inverse Sine Function
For a function to have an inverse function, it must be one-to-one—that
is, it must pass the Horizontal Line Test.
sin x has an inverse function
on this interval.
Restrict the domain to the interval – / 2 x
/ 2, the following properties hold.
1. On the interval [– / 2, / 2], the function
y = sin x is
increasing.
2. On the interval [– / 2, / 2], y = sin x
takes on its full
range of values, –1 sin x 1.
3. On the interval [– / 2, / 2], y = sin x is
one-to-one.
By definition, the values of inverse
trigonometric functions are always
in radians.
5
Inverse Sine Function
On the restricted domain – / 2 x / 2, y = sin x has a
unique inverse function called the inverse sine function.
y = arcsin x or y = sin –1 x.
means the angle (or arc) whose sine is x.
6
Inverse Sine Function
7
Example 1 – Evaluating the Inverse Sine Function
If possible, find the exact value.
a. b. c.
Solution:
a. Because for ,
. Angle whose sine is
8
Example 1 – Solution
b. Because for ,
.
c. It is not possible to evaluate y = sin –1 x when x = 2
because there is no angle whose sine is 2.
Remember that the domain of the inverse sine function
is [–1, 1].
cont’d
Angle whose sine is
9
Example 2 – Graphing the Arcsine Function
Sketch a graph of y = arcsin x.
Solution:
In the interval [– / 2, / 2],
10
Example 2 – Solution
y = arcsin x
Domain: [–1, 1]
Range: [– / 2, / 2]
cont’d
11
Other Inverse Trigonometric
Functions
12
Other Inverse Trigonometric Functions
The cosine function is decreasing and one-to-one on the
interval 0 x .
On the interval 0 x the cosine function has an inverse
function—the inverse cosine function:
y = arccos x or y = cos –1 x.
cos x has an inverse function on this interval.
13
Other Inverse Trigonometric Functions
DOMAIN: [–1,1]
RANGE:
DOMAIN: [–1,1]
RANGE: [0, ]
DOMAIN:
RANGE:
14
Example 3 – Evaluating Inverse Trigonometric Functions
Find the exact value.
a. arccos b. cos –1(–1)
c. arctan 0 d. tan –1 (–1)
Solution:
a. Because cos ( / 4) = , and / 4 lies in [0, ],
. Angle whose cosine is
15
Example 3 – Solution
b. Because cos = –1, and lies in [0, ],
cos –1(–1) = .
c. Because tan 0 = 0, and 0 lies in (– / 2, / 2),
arctan 0 = 0.
Angle whose cosine is –1
Angle whose tangent is 0
cont’d
16
Example 3 – Solution
d. Because tan(– / 4) = –1, and – / 4 lies in (– / 2, / 2),
tan –1 (–1) = . Angle whose tangent is –1
cont’d
17
Compositions of Functions
18
Compositions of Functions
For all x in the domains of f and f –1, inverse functions have the
properties
f (f –1(x)) = x and f
–1 (f (x)) = x.
19
Compositions of Functions
These inverse properties do not apply for arbitrary values
of x and y.
.
The property arcsin(sin y) = y
is not valid for values of y outside the interval [– / 2, / 2].
20
Example 5 – Using Inverse Properties
If possible, find the exact value.
a. tan[arctan(–5)] b. c. cos(cos –1 )
Solution:
a. Because –5 lies in the domain of the arctan function, the
inverse property applies, and you have
tan[arctan(–5)] = –5.
21
Example 5 – Solution
b. In this case, 5 / 3 does not lie within the range of the
arcsine function, – / 2 y / 2.
However, 5 / 3 is coterminal with
which does lie in the range of the arcsine function,
and you have
cont’d
22
Example 5 – Solution
c. The expression cos(cos –1) is not defined because
cos –1 is not defined.
Remember that the domain of the inverse cosine
function is [–1, 1].
cont’d
23
Find the exact value: tan arccos2
3
Let 𝑢 = arccos2
3, then cos 𝑢 =
2
3> 0, therefore, u is in QI
tan arccos2
3= tan𝑢 =
sin 𝑢
cos 𝑢=
1 −23
2
23
=5
2
Example 6 – Evaluating Compositions of Functions
24
Find the exact value: cos arc𝑠𝑖𝑛(−3
5)
Let 𝑢 = arc𝑠𝑖𝑛(−3
5), then sin 𝑢 = −
3
5< 0, therefore, u is in
Q IV
cos arc𝑠𝑖𝑛(−3
5) = 1 − −
3
5
2
=4
5
Example 6 – Evaluating Compositions of Functions
25
Example
Write the expression as an algebraic expression in x:
sin arccos 3𝑥 , 0 ≤ 𝑥 ≤1
3
sin arccos 3𝑥 = 1 − 9𝑥2