1. Section 5.5 Inverse Trigonometric Functions & Their Graphs
Chapter 5 – Trigonometric Functions: Unit Circle Approach
Section Inverse Trigonometric Functions & Their Graphs
5.5 - Inverse Trigonometric Functions & Their Graphs

2. Review of Inverse Functions
Remember
If the graph passes the horizontal line test, then the function has an inverse functions.
If a point (a, b) is on the graph of f, then the point (b, a) is on the graph of f -1.
The graph of f -1 is a reflection of the graph of f about the line y=x.
5.5 - Inverse Trigonometric Functions & Their Graphs

3. 5.5 - Inverse Trigonometric Functions & Their Graphs
Sine Function
Does not pass the horizontal line test.
Must restrict the domain to create an inverse function.
5.5 - Inverse Trigonometric Functions & Their Graphs

4. 5.5 - Inverse Trigonometric Functions & Their Graphs
Definition
The inverse sine function is the function sin-1 with domain [-1, 1] and range [- ⁄ 2,  ⁄ 2] defined by
The inverse sine function is also called arcsine denoted by arcsin.
5.5 - Inverse Trigonometric Functions & Their Graphs

5. 5.5 - Inverse Trigonometric Functions & Their Graphs
Note
5.5 - Inverse Trigonometric Functions & Their Graphs

6. 5.5 - Inverse Trigonometric Functions & Their Graphs
Graph of Inverse sine
5.5 - Inverse Trigonometric Functions & Their Graphs

7. Cancellation Properties - Sine
Thus y = sin-1x is the number in the interval [- ⁄ 2,  ⁄ 2] whose sine is x.
In other words we have the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

8. 5.5 - Inverse Trigonometric Functions & Their Graphs
Examples
Find the exact value of the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

9. 5.5 - Inverse Trigonometric Functions & Their Graphs
Cosine Function
Does not pass the horizontal line test.
Must restrict the domain to create an inverse function.
5.5 - Inverse Trigonometric Functions & Their Graphs

10. 5.5 - Inverse Trigonometric Functions & Their Graphs
Definition
The inverse cosine function is the function cos-1 with domain [-1, 1] and range [0, ] defined by
The inverse sine function is also called arccosine denoted by arccos.
5.5 - Inverse Trigonometric Functions & Their Graphs

11. Graph of Inverse Cosine
5.5 - Inverse Trigonometric Functions & Their Graphs

12. Cancellation Properties - Cosine
Thus y = cos-1x is the number in the interval [0, ] whose cosine is x.
In other words we have the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

13. 5.5 - Inverse Trigonometric Functions & Their Graphs
Examples
Find the exact value of the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

14. 5.5 - Inverse Trigonometric Functions & Their Graphs
Tangent Function
Does not pass the horizontal line test.
Must restrict the domain to create an inverse function.
5.5 - Inverse Trigonometric Functions & Their Graphs

15. 5.5 - Inverse Trigonometric Functions & Their Graphs
Definition
The inverse tangent function is the function tan-1 with domain (-∞, ∞) and range (- ⁄ 2,  ⁄ 2) defined by
The inverse tangent function is also called arctangent denoted by arctan.
5.5 - Inverse Trigonometric Functions & Their Graphs

16. Graph of Inverse Tangent
5.5 - Inverse Trigonometric Functions & Their Graphs

17. Cancellation Properties - Tangent
Thus y = tan-1x is the number in the interval (- ⁄ 2,  ⁄ 2) whose sine is x.
In other words we have the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

18. 5.5 - Inverse Trigonometric Functions & Their Graphs
Examples
Find the exact value of the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

19. Evaluating Compositions
5.5 - Inverse Trigonometric Functions & Their Graphs

20. 5.5 - Inverse Trigonometric Functions & Their Graphs
Inverse Properties
5.5 - Inverse Trigonometric Functions & Their Graphs

21. Using Inverse Properties
Evaluate the following:
5.5 - Inverse Trigonometric Functions & Their Graphs

22. 5.5 - Inverse Trigonometric Functions & Their Graphs
Examples – pg. 412
Find the exact value of the expression if it is defined.
5.5 - Inverse Trigonometric Functions & Their Graphs