1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 4
Trigonometric Functions
4.2 Trigonometric
Functions: The Unit Circle
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Use a unit circle to define trigonometric functions of real numbers.
Recognize the domain and range of sine and cosine functions.
Find exact values of the trigonometric functions at
Use even and odd trigonometric functions.
Recognize and use fundamental identities.
Use periodic properties.
Evaluate trigonometric functions with a calculator.

3. The Unit Circle
A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is

4. The Unit Circle (continued)
In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t.

5. The Six Trigonometric Functions

6. Definitions of the Trigonometric Functions in Terms of a Unit Circle

7. Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.

8. Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.

9. The Domain and Range of the Sine and Cosine Functions

10. Exact Values of the Trigonometric Functions at
Trigonometric functions at occur frequently. We use the unit circle to find values of the trigonometric functions at
We see that point P = (a, b)
lies on the line y = x. Thus, point P
has equal x- and y-coordinates: a = b.

11. Exact Values of the Trigonometric Functions at (continued)
We find these coordinates as follows:

12. Exact Values of the Trigonometric Functions at (continued)
We have used the unit circle to find the coordinates of point P = (a, b) that correspond to

13. Example: Finding Values of the Trigonometric Functionsat
Find
Find

14. Example: Finding Values of the Trigonometric Functionsat
Find

15. Trigonometric Functions at

16. Even and Odd Trigonometric Functions

17. Example: Using Even and Odd Functions to Find Values of Trigonometric Functions
Find the value of each trigonometric function:

18. Fundamental Identities

19. Example: Using Quotient and Reciprocal Identities
Given and find the value of each of the four remaining trigonometric functions.

20. Example: Using Quotient and Reciprocal Identities (continued)
Given and find the value of each of the four remaining trigonometric functions.

21. The Pythagorean Identities

22. Example: Using a Pythagorean Identity
Given that and find the value of
using a trigonometric identity.

23. Definition of a Periodic Function

24. Periodic Properties of the Sine and Cosine Functions

25. Periodic Properties of the Tangent and Cotangent Functions

26. Example: Using Periodic Properties
Find the value of each trigonometric function:

27. Repetitive Behavior of the Sine, Cosine, and Tangent Functions

28. Using a Calculator to Evaluate Trigonometric Functions
To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions.

29. Example: Evaluating Trigonometric Functions with a Calculator
Use a calculator to find the value to four decimal places: