1. Inverse Trigonometric Functions
Section 4.5
Inverse Trigonometric Functions

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

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-1 x or y = arcsin x. This notation means “y is the angle whose sine is x.”

5. The notation sin-1 indicates an angle!

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

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

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

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

10. EXAMPLE 1 y = Sin-1 1. Find y.
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 -

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

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

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

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

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

16. EXAMPLE 4 Find Cos-1(- ) 2 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  = . 2  4 3

17. Homework pp

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

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

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

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

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

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

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

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

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

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

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