Inverse Trigonometric Functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

Inverse Trigonometric Functions

Inverse Trig Functions Objectives

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

• Evaluate inverse sine, cosine, and tangent functions.

• Evaluate compositions of inverse trig functions.

• Use inverse trig functions in applications.

Definitions and Terminology


-1

1

sin x or arcsinx is the angle whose sine is x.

sin sinx x

-1

1

cos x or arccosx is the angle whose cosine is x.

cos cosx x

-1

1

tan x or arctanx is the angle whose tan is x.

tan tanx x


Restricting the Sine Function

y

2

1

x

y = sin x

Sin x has an inverse function on this interval.

Recall that for a function to have an inverse on its entire domain, 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 and must be restricted to for its inverse to be a function.


The inverse sine function is defined byy = arcsin x if and only if sin y = x.

Angle whose sine is x

The domain of y = arcsin x is [–1, 1].

Examples:

1a. arcsin2

1 3b. sin2

The range of y = arcsin x is [–/2 , /2].

3

6 1 is the angle whose sine is .

6 2

3sin3 2

(0,–1)

(0,1)

xx


Restricting the Cosine Function

Cos x has an inverse function on this interval.

f(x) = cos x must be restricted to find its inverse.

y

2

1

x

y = cos x

pair/share


The inverse cosine function is defined byy = arccos x if and only if cos y = x.

Angle whose cosine is x

The domain of y = arccos x is [–1, 1].

Examples:

1a.) arccos2

1 is the angle whose cosine is .3 2

1 3b.) cos2

35cos6 2

The range of y = arccos x is [0 , ].

3

56

(0,1)

y

(0,-1)


The inverse tangent function is defined byy = arctan x if and only if tan y = x.

Angle whose tangent is x

Example:

3a.) arctan3

3 is the angle whose tangent is .6 3

1b.) tan 1 tan 14

The domain of y = arctan x is .( , ) The range of y = arctan x is [–/2 , /2].

(0,–1)

(0,1)

xx

4

6


Graphing Utility: Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

–1.5 1.5

–

–1.5 1.5

2

–

–3 3

–

Set calculator to radian mode.


Graphing Utility: Approximate the value of each expression.

a. cos–1 0.75 b. arcsin 0.19

c. arctan 1.32 d. arcsin 2.5

Set calculator to radian mode.


If –1 x 1, then sin(arcsin x) = x and if – /2 y /2, then arcsin(sin y) = y.

If –1 x 1, then cos(arccos x) = x and if 0 y , then arccos(cos y) = y.

If x is a real number, then tan(arctan x) = x and if –/2 < y < /2, then arctan(tan y) = y.

Inverse Properties:


Examples:

1 4cos cos

3

2

3

tan arctan 4 4


You try:

a. sin–1(sin (–/2)) =

1 5b. sin sin3

53 does not lie in the range of the arcsine function, –/2 y /2.

y

x

53

3

1 5sin sin3 3

–/2


Evaluating Composition of Functions

2Find the exact value of tan arccos .3

x

y

3

2

5

opp 52tan arccos tan3 adj 2

u

u

Precalculus 4.7 Inverse Trigonometric Functions

21

3Find the exact value of: cos arcsin

5

x

y

You Try:


Write each of the following as an algebraic expression in x:

1) sin arccos3 03

a x x

x

y

2) 1 9a x

1) cot arccos3 03

b x x

2

3)1 9

xbx

120 ft

H = 74.98 ft

A person walks 120 ft. away from a building. The line of sight to the top of the building is 150 ft. What is the angle of elevation to the top of the building?

Applying Inverse Trig Functions:

36.87


A person stands 50 ft. from a tree. If the height of the tree is 70 ft., find the angle of elevation to the top of the tree.

54.46

Homework

4.7 p 316:

1-7 odd, 13-53 odd


Inverse Trigonometric Functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

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