1. 4.7 Inverse Trig Functions

2. Inverse trig functions
What trig functions can we evaluate without using a calculator?
Sin 𝜋 4
Cos 𝜋 3
Tan 𝜋 6
Sin 𝜋 2

3. Inverse Trig Functions
What does an inverse function do?
Finds the input of a function when given the output
How can we determine if a function has an inverse?
Horizontal Line Test
If any horizontal line intersects the graph of a function in more than one point, the function does not have an inverse

4. Does the Sine function have an inverse?1 -1

5. What could we restrict the domain to so that the sine function does have an inverse?1 -1

6. Inverse Sine, , arcsine (x)
Function is increasing
Takes on full range of values
Function is 1-1
Domain:
Range:

7. Evaluate: arcSin
Asking the sine of what angle is

8. Find the following:
ArcSin

9. Inverse Cosine Function
What can we restrict the domain of the cosine curve to so that it is 1-1? 1 -1

10. Inverse Cosine, , arcCos (x)
Function is increasing
Takes on full range of values
Function is 1-1
Domain:
Range:

11. Evaluate: ArcCos (-1)
The Cosine of what angle is -1?

12. Evaluate the following:
ArcCos

13. arcCos (0.28) Is the value 0.28 on either triangle or curve?
Use your calculator:

14. Determine the missing Coordinate

15. Determine the missing Coordinate

16. Use an inverse trig function to write θ as a function of x.

17. Find the exact value of the expression.
Sin [ ArcCos ]

18. 4.7 Inverse Trig Functions

20. So far we have:
Restricted the domain of trig functions to find their inverse
Evaluated inverse trig functions for exact values

21. ArcTan (x) Similar to the ArcSin (x) Domain of Tan Function:
Range of Tan Function:

22. Composition of Functions
From Algebra II: If two functions, f(x) and f −1 (x), are inverses, then their compositions are: f( f −1 (x)) = x and f −1 (f(x)) = x

23. Inverse Properties of Trig Functions
If -1 ≤ x ≤ 1 and - 𝜋 2 ≤ y ≤ 𝜋 2 , then
Sin (arcSin x) = x and arcSin (Sin y) = y
If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then
Cos (arcCos x) = x and arcCos (Cos y) = y
If x is a real number and - 𝜋 2 < y < 𝜋 2 , then
Tan (arcTan x) = x and arcTan (Tan y) = y

24. Inverse Trig Functions
Use the properties to evaluate the following expression:
Sin (ArcSin 0.3)

25. Inverse Trig Functions
Use the properties to evaluate the following expression:
ArcCos (Cos 2𝜋 3 )

26. Inverse Trig Functions
Use the properties to evaluate the following expression:
ArcSin (Sin 3π)

27. Inverse Trig Functions
Use the properties to evaluate the following expression:
Tan (ArcTan 25)
Cos (ArcCos -0.2)
ArcCos (Cos 7𝜋 2 )

28. 4.7 Inverse Trig Functions

29. Inverse Trig Functions
Yesterday, we only had compositions of functions that were inverses
When we have a composition of two functions that are not inverses, we cannot use the properties
In these cases, we will draw a triangle

30. Inverse Trig Functions
Sin (arcTan )
Let u = whatever is in parentheses
u = arcTan 3 4
→ Tan u = 3 4

31. Inverse Trig Functions
Sec (arcSin )

32. Inverse Trig Functions
Sec (arcSin )
Cot (arcTan )
Sin (arcTan x)

33. Inverse Trig Functions
In this section, we have:
Defined our inverse trig functions for specific domains and ranges
Evaluated inverse trig functions
Evaluated compositions of trig functions
2 Functions that are inverses
2 Functions that are not inverses by evaluating the inner most function first
2 Functions that are not inverses by drawing a triangle

34. Sine Function 1
- 𝜋 2
𝜋 2 -1

35. Cosine Function 1 π
𝜋 2 -1

36. Tangent Function
- 𝜋 2
𝜋 2

37. Evaluating Inverse Trig Functions
arcTan ( )
Cos −1 (− )
arcSin (-1)

38. Composition of Functions
When the two functions are inverses:
Sin (arcSin -0.35)
arcCos (Cos 3𝜋 4 )

39. Composition of Functions
When the two functions are not inverses:
Sin −1 (Cos 11π 6 )
arcTan (Sin 4𝜋 3 )

40. Composition of Functions
When the two functions are not inverses:
Sin (arcCos )
Cot ( Sin − )

41. 4.7 Inverse Trig Functions

42. Inverse Trig Functions
Evaluate the following function:
f(x) = Sin (arcTan 2x)
In your graphing calculator, graph both of these functions.

43. Inverse Trig Functions
Solve the following equation for the missing piece:
arcTan 9 x = arcSin (___)

44. Inverse Trig Functions
Find the missing pieces in the following equations:
arcSin −x = arcCos (___)
arcCos x 2 −2x = arcSin (___)
arcCos x −2 2 = arcTan (___)

45. Inverse Trig Functions

46. Inverse Trig Functions

47. Composition of Functions
Evaluate innermost function first
Substitute in that value
Evaluate outermost function

48. Sin (arcCos ) Evaluate the innermost function first: arcCos ½ =
Substitute that value in original problem

50. How do we evaluate this?
Let θ equal what is in parentheses

51. 13 12 θ 5

52. 13 12
How do we evaluate this? θ 5
Let θ equal what is in parentheses
Use the triangle to answer the question

54. What is different about this problem?
Is 0.2 in the domain of the arcSin?

55. What is different about this problem?

56. Graph of the ArcSin Y
X = Sin Y

57. Graph of the ArcSin

58. Graph of ArcCos Y
X = Sin Y

59. Graph of the ArcCos