Inverse Trigonometric Functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 HWQ Write a sine equation that has an amplitude of 2, a period of and a y-intercept of -1. Where does your function equal 0?

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Where does your function equal 0?

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

6 What are you being asked to find? Let’s look at the graph!

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Inverse Sine Function Restricting the Sine Function y 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.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Inverse Sine Function The inverse sine function is defined by y = arcsin x if and only ifsin y = x. Angle whose sine is x The domain of y = arcsin x is [–1, 1]. Examples: The range of y = arcsin x is [–  /2,  /2]. (0,–1) (0,1) x x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Inverse Cosine Function 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 x y = cos x pair/share

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Inverse Cosine Function The inverse cosine function is defined by y = arccos x if and only ifcos y = x. Angle whose cosine is x The domain of y = arccos x is [–1, 1]. Examples: The range of y = arccos x is [0,  ]. (0,1) y (0,-1)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. Tan x has an inverse function on this interval. y x y = tan x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only iftan y = x. Angle whose tangent is x Example: The domain of y = arctan x is. The range of y = arctan x is [–  /2,  /2]. (0,–1) (0,1) x x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Graphing Utility: Graphs of Inverse Functions Graphing Utility: Graph the following inverse functions. a. y = arcsin x b. y = arccos x c. y = arctan x – ––  – 2 –– –3 3  –– Set calculator to radian mode.

Precalculus 4.7 Inverse Trigonometric Functions 14 Find the exact values. You try:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. a. cos – b. arcsin 0.19 c. arctan 1.32d. arcsin 2.5 Set calculator to radian mode.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Composition of Functions Using Properties of Inverse Functions: f (f –1 (x)) = x only if x lies in the domain of f –1 f –1 (f (x)) = x only if x lies in the range of f –1 Examples:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Composition of Functions 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:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Composition of Functions Examples:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Composition of Functions You try: a. sin –1 (sin (–  /2)) = does not lie in the range of the arcsine function, –  /2  y   /2. y x –  /2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Example: Evaluating Composition of Functions Evaluating Composition of Functions x y 3 2 u

Precalculus 4.7 Inverse Trigonometric Functions 21  x y You Try:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Example: Evaluating Composition of Functions Write each of the following as an algebraic expression in x: x y

 120 ft H = 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:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 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. 

Lesson Check Solve the equation on 2sin(x)+1=0 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25

Homework 4.7 p 316: 1-7 odd, odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26