1. Inverse Trigonometric Functions Digital Lesson

2. 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. 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]. The sine of what angle is = to √3/2?

4. More examples Arcsin 0 4 Sin -1 (-√2/2) Arcsin 2

5. Graph the arcsin x First, rewrite as x = sin y 5 Table x y -  /2 -  /4 0  /4  /2  /4  /2  /4

6. 6 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

7. 7 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,  ].

8. More examples Arccos 0 8 cos -1 (-√2/2) Arccos 2

9. Graph the arccos x First, rewrite as x = cos y 9 Table x y 0  /4  /2 3  /4   /2  /4 

10. 10 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

11. 11 Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only iftan y = x. Angle whose tangent is x The domain of y = arctan x is. The range of y = arctan x is [–  /2,  /2].

12. More examples Arctan 0 12 tan -1 (-√3/3) Arctan 1

13. 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 –1.5 1.5 ––  –1.5 1.5 22 –– –3 3  –– Set calculator to radian mode.

14. 14 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.

15. 15 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:

16. 16 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.

17. 17 Example: Evaluating Composition of Functions Example: x y 3 2 u

18. 18 Example: Evaluating Composition of Functions Example: x y 3 2 u

19. 19 Example: Evaluating Composition of Functions Example: x y -5 12 u Find the exact value of csc[arc tan(-5/12)] Let u = arc tan (-5/12), then tan u = opp = -5 adj 12 12 2 + (-5) 2 = c 2 c = 13 Csc u = hyp = 13 opp -5

20. 20 Example: Evaluating Composition of Functions Example: x y x 1 u Find the exact value of sin[arc tan(x)] Let u = arc tan (x), then tan u = opp = x adj 1 x 2 + (1) 2 = c 2 c =  x 2 + 1 sin u = opp = x hyp  x 2 + 1