Section 4.5 Inverse Trigonometric Functions

1. Section 4.5 Inverse Trigonometric Functions

2. Objectives: 1. To define inverse trigonometric functions. 2. To evaluate inverse trigonometric functions. 3. To graph inverse trigonometric functions.

3. Remember, to find the rule for an inverse function, you interchange the x and y and solve for y.

4. In the case of y = sin x, solving x = sin y for y requires a symbol for the inverse. Mathematicians use the notation y = sin-1x or y = arcsin x. This notation means “y is the angle whose sine is x.”

5. The notation sin-1 indicates an angle!

6. 1 -2 -  2 -1 Consider the sin function. Since the function is not one-to-one the inverse is not a function.

7.  2 1 -2 -  2 -1 - 2 To make the original function one-to-one we restrict the domain of sin to [ , ]

8. 1 -2 -  2 -1 This is the graph of f(x) = Sin x.

9. 1 -2 -  2 -1 By reflecting Sin x across the line y = x we get the graph of f(x) = Sin-1 x.

10. y = Sin-1 1 is an equivalent expression to sin y = 1. In other words, we want to know the angle whose sin is 1. Since the “s” in sin is capitalized we want the angle from the restricted domain [ , ].  2 - 2 EXAMPLE 1y = Sin-1 1. Find y.

11.  2 y = EXAMPLE 1y = Sin-1 1. Find y. y = Sin-1 1

12. EXAMPLE 2Find sin(Sin-1 ) sin(Sin-1 ) = sin =  6 1 2 1 2 1 2

13. EXAMPLE 3Find sin(Cos-1 ) 4 3 4 x  3 Cos-13/4 represents an angle in [0, ]. Since 3/4 is positive it is a first quadrant angle. Therefore you have the following right triangle.

14. EXAMPLE 3Find sin(Cos-1 ) 4 3 4 x  3 x = 7 Use the Pythagorean theorem to find the missing side. 32 + x2 = 42 9 + x2 = 16 x2 = 7

15. EXAMPLE 3Find sin(Cos-1 ) 4 7 3 4 3 4  3 7 4 ∴ sin(Cos-1 ) =

16. EXAMPLE 4Find Cos-1(- ) Since the cosine is negative, the angle is in the second quadrant. The cos = . The angle in the second quadrant with a reference angle of is the angle  - = .  4  4 2 2 2 2 2 2 3 4

17. Homework pp. 196-197

18. ►A. Exercises Graph. 1. y = cos x 1 

19. ►A. Exercises Graph. 2. y = Cos x, x  [0, ] 1 

20. ►A. Exercises Graph. 3. y = Cos-1 x  1  1

21. ►A. Exercises Graph. 4. y = tan x 1 

22.  2 ►A. Exercises Graph. 5. y = Tan x, x  ( , ) - 2 1 

23. ►A. Exercises Graph. 6. y = Tan-1 x

24. 3 2 ►A. Exercises Without using a calculator, find the following values. 7. Sin-1

25. 1 2 ►A. Exercises Without using a calculator, find the following values. 13. tan(Sin-1 )

26. ►A. Exercises Without using a calculator, find the following values. 15. cos(Sin-1 ) - 5 3

27. ►A. Exercises Use a calculator to determine the following values. 17. Sin-1 0.3420

28. ►B. Exercises Graph the given function over its appropriate restricted domain. (State the restricted domain.) Graph its inverse function on the same set of axes. 21. g(x) = Csc x

29. ►B. Exercises Use the definitions and a calculator to evaluate the following. 23. Cot-1 0.684

30. ►B. Exercises Use the definitions and a calculator to evaluate the following. 27. Sin-1 0.7854

31. ■ Cumulative Review 35. Give the angle of inclination of the line 3x + 4y = 7 to the nearest degree.

32. ■ Cumulative Review 36. Change f(x) = x – to general form. 5 7 1 4

33. ■ Cumulative Review 37. Give the function rule for the line passing through the points (-4, 5), (3, 8.5), and (8, 11).

34. ■ Cumulative Review 38. Find an equivalent expression for f(x) = sec ( – x).  2

35. ■ Cumulative Review 39. Find the inverse of the function f(x) = x – 5. 2 3

36. 1 -2 -  2 -1 y = sin x y = csc x

37. 1 -2 -  2 -1 y = cos x y = sec x

38. 1 -2 -  2 -1 y = tan x y = cot x