4.7 INVERSE TRIGONOMETRIC FUNCTIONS

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• 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

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Inverse Sine Function

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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.

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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.

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Inverse Sine Function

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Example 1 – Evaluating the Inverse Sine Function

If possible, find the exact value.

a. b. c.

Solution:

a. Because for ,

. Angle whose sine is

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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

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Example 2 – Graphing the Arcsine Function

Sketch a graph of y = arcsin x.

Solution:

In the interval [– / 2, / 2],

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Example 2 – Solution

y = arcsin x

Domain: [–1, 1]

Range: [– / 2, / 2]

cont’d

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Other Inverse Trigonometric

Functions

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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.

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Other Inverse Trigonometric Functions

DOMAIN: [–1,1]

RANGE:

DOMAIN: [–1,1]

RANGE: [0, ]

DOMAIN:

RANGE:

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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

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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

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Example 3 – Solution

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

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

cont’d

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Compositions of Functions

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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.

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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].

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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.

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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

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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

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Find the exact value: tan arccos2

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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

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=5

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Example 6 – Evaluating Compositions of Functions

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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

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Example 6 – Evaluating Compositions of Functions

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Example

Write the expression as an algebraic expression in x:

sin arccos 3𝑥 , 0 ≤ 𝑥 ≤1

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sin arccos 3𝑥 = 1 − 9𝑥2