Digital Lesson Inverse Trigonometric Functions Inverse Sine Function

Digital Lesson Inverse Trigonometric Functions

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 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. 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 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. 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 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

Example: y 3 u 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x 8

Deriving Inverse Trig Fuctions Tips to remember: If it starts with ‘c’, make it negative! 9

Deriving Inverse Trig Fuctions 10

Deriving Inverse Trig Fuctions 11

Another way to write inverse functions 12

Spotlight Search… Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

…Using inverse trig functions: