Digital Lesson Inverse Trigonometric Functions Inverse Sine Function

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Digital Lesson Inverse Trigonometric Functions

Digital Lesson Inverse Trigonometric Functions

Inverse Sine Function Recall that for a function to have an inverse, it must

Inverse Sine Function Recall that for a function to have an inverse, 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 find its inverse. y y = sin x x Sin x has an inverse function on this interval. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

The inverse sine function is defined by y = arcsin x if and only

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 [– /2 , /2]. Example: This is another way to write arcsin x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

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

Inverse Cosine Function f(x) = cos x must be restricted to find its inverse. y y = cos x x Cos x has an inverse function on this interval. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

The inverse cosine function is defined by y = arccos x if and only

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 , ]. Example: This is another way to write arccos x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

Inverse Tangent Function f(x) = tan x must be restricted to find its inverse.

Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. y y = tan x x Tan x has an inverse function on this interval. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

The inverse tangent function is defined by y = arctan x if and only

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 [– /2 , /2]. Example: This is another way to write arctan x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Graphing Utility: Graph the following inverse functions. Set calculator to radian mode. a. y

Graphing Utility: Graph the following inverse functions. Set calculator to radian mode. a. y = arcsin x – 1. 5 – 2 b. y = arccos x – 1. 5 – c. y = arctan x – 3 3 – Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Graphing Utility: Approximate the value of each expression. Set calculator to radian mode. a.

Graphing Utility: Approximate the value of each expression. Set calculator to radian mode. a. cos– 1 0. 75 b. arcsin 0. 19 c. arctan 1. 32 d. arcsin 2. 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

Composition of Functions: f(f – 1(x)) = x and (f – 1(f(x)) = x.

Composition of Functions: f(f – 1(x)) = x and (f – 1(f(x)) = x. Inverse Properties: If – 1 x 1 and – /2 y /2, then sin(arcsin x) = x and arcsin(sin y) = y. If – 1 x 1 and 0 y , then cos(arccos x) = x and arccos(cos y) = y. If x is a real number and – /2 < y < /2, then tan(arctan x) = x and arctan(tan y) = y. Example: tan(arctan 4) = 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

Example: a. sin– 1(sin (– /2)) = – /2 does not lie in the

Example: a. sin– 1(sin (– /2)) = – /2 does not lie in the range of the arcsine function, – /2 y /2. y However, it is coterminal with which does lie in the range of the arcsine x function. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

Example: y 3 u 2 Copyright © by Houghton Mifflin Company, Inc. All rights

Example: y 3 u 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 12