1. Inverse Trigonometric Functions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Inverse 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, 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.

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

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Inverse 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

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 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]. Example: This is another way to write arccos x. The range of y = arccos x is [0,  ].

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 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: This is another way to write arctan x. The domain of y = arctan x is. The range of y = arctan x is [–  /2,  /2].

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 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 –1.5 1.5 ––  –1.5 1.5 22 –– –3 3  –– Set calculator to radian mode.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. a. cos – 1 0.75b. arcsin 0.19 c. arctan 1.32d. arcsin 2.5 Set calculator to radian mode.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Composition of Functions Composition of Functions: f(f –1 (x)) = x and (f –1 (f(x)) = x. 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 Inverse Properties:

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

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