# 4_7 INVERSE TRIG FNS.pdf

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• 4.7 INVERSE TRIGONOMETRIC FUNCTIONS

• 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-onethat 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

• 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].

contd

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]

contd

• 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 functionthe 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

contd

• 16

Example 3 Solution

d. Because tan( / 4) = 1, and / 4 lies in ( / 2, / 2),

tan 1 (1) = . Angle whose tangent is 1

contd

• 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

contd

• 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].

contd

• 23

Find the exact value: tan arccos2

3

Let = arccos2

3, then cos =

2

3> 0, therefore, u is in QI

tan arccos2

3= tan =

sin

cos =

1 23

2

23

=5

2

Example 6 Evaluating Compositions of Functions

• 24

Find the exact value: cos arc(3

5)

Let = arc(3

5), then sin =

3

5< 0, therefore, u is in

Q IV

cos arc(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 3 , 0 1

3

sin arccos 3 = 1 92