416 Inverse Trigonometric Functions EQ What are inverse

4/16 Inverse Trigonometric Functions EQ: What are inverse trigonometric functions and how do I find domain, range, and exact values using them?

Quick Review • How do you find inverses of functions? - Switch x & y - Solve for y. • Are inverses of functions always functions? – How did we test for this? - No - Use the vertical line test (x cannot repeat)

Inverse Trig Functions Original Function Inverse y = sin x y = sin-1 x y = arcsin x y = cos-1 x y = arccos x y = tan-1 x y = arctan x

Consider the graph of y = sin x What is the domain and range of sin x? D: (-∞, ∞) R: [-1, 1] What would the graph of y = arcsin x look like? What is the domain and range of arcsin x? D: [-1, 1] R: (-∞, ∞)

Is the inverse of sin x a function? • This will also be true for cosine and tangent. • Therefore all of the domains are restricted in order for the inverses to be functions.

How do you know if the domain is restricted for the original functions? • Capital letters are used to distinguish when the function’s domain is restricted. Original Functions with Restricted Domain Inverse Function y = Sin x y = Sin-1 x y = Arcsin x y = Cos-1 x y = Arccos x y = Tan-1 x y = Arctan x

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. 7

The inverse sine function is defined by y = arcsin x if and only if sin y = x. Angle whose sine is x [-1, 1] The domain of y = Arcsin x is _______. [-л/2 , л/2] The range of y = Arcsin x is ____. Example: This is another way to write arcsin x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

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. 9

The inverse cosine function is defined by y = arccos x if and only if cos y = x. Angle whose cosine is x [-1, 1] The domain of y = Arccos x is _______. [0, л] The range of y = Arccos x is ____. Example: This is another way to write arccos x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

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. 11

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 ____. (-л/2 , л/2) Example: This is another way to write arctan x. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

Question! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

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. 14

Graphing Utility: Approximate the value of each expression. Set calculator to radian mode. Round three decimal places. a. cos– 1 0. 75 0. 723 c. arctan 1. 32 0. 922 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. b. arcsin 0. 191 d. arccos 2. 5 undefined 15

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) = ____ Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Example: a. sin– 1(sin (– /2)) = _____ y x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

Example: y u Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 18

Example: u y x 2 + y 2 = h 2 22 + y 2 = 32 y 2 4+ =9 y 2 = 5 y = sqrt(5) u Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 19

You Try: Evaluate each expression.

You Try: Evaluate each expression.