Inverse Trigonometric Functions

Graph y = sinx Graph y = -1/2 Domain: x is element of all reals Range: {y| -1 ≤ y ≤ 1} Graph y = -1/2 Where do these graphs cross? Is y = sinx one-to-one?

Inverse Sine Function Sin x has an inverse function on this interval. 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 is not one-to-one and must be restricted to find its inverse. y x y = sin x Sin x has an inverse function on this interval. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Sine Function

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 {x | –1≤ x ≤ 1} The range of y = Arcsin x is { y | –/2 ≤ y ≤ /2} Example: This is another way to write Arcsin x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Sine Function

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

Inverse Cosine Function 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 { x | –1 ≤ x ≤ 1} The range of y = Arccos x is { y | 0 ≤ y ≤ } Example: This is another way to write Arccos x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Cosine Function

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

Inverse Tangent Function 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 all real numbers The range of y = Arctan x is { y | –/2 ≤ y ≤ /2} Example: This is another way to write Arctan x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Tangent Function

Graphing Utility: Graphs of Inverse Functions Graphing Utility: Graph the following inverse functions. Set calculator to radian mode. –1.5 1.5 –  a. y = Arcsin x –1.5 1.5 2 – b. y = Arccos x –3 3  – c. y = Arctan x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphing Utility: Graphs of Inverse Functions

Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. Set calculator to radian mode. a. Cos–1 0.75 b. Arcsin 0.19 Why? c. Arctan 1.32 d. Arcsin 2.5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphing Utility: Inverse Functions

Composition of Functions Recall  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. Composition of Functions

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

Example: Evaluating Composition of Functions y 3 u 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Evaluating Composition of Functions

Example: Evaluating Composition of Functions OR… y 2 x 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Evaluating Composition of Functions