1. Chapter 4 Trigonometric Functions 1

2. 4.4 Trig Functions of Any Angle Objectives:  Evaluate trigonometric functions of any angle.  Use reference angles to evaluate trigonometric functions.  Evaluate trigonometric functions of real numbers. 2

3. Brief Review  Trig functions defined using the unit circle:  sin θ =  cos θ =  tan θ = 3

4. Trig Functions of Any Angle  Let θ be an angle in standard position with (x, y) a point on the terminal side of θ.  Create a right triangle and define the six trig functions. 4

5. Trig Functions of Any Angle  The six trig functions are defined:  Note: r 2 = x 2 + y 2 or  r is always positive. 5

6. Example 1  Let (12, –5) be a point on the terminal side of θ. Find the six trig functions of θ. 6

7. Signs of Trig Functions  All of the trig functions are positive (+) in Quadrant I because both x and y are positive.  The signs of the trig functions in the other quadrants are dependent on the signs of x and y in those quadrants. 7 AllStudents Take Calc

8. Example 2  Given cos θ = –4/5 and sin θ > 0, find the six trig functions of θ. 8

9. Trig Functions of Quadrant Angles  How do we find the trig functions of the quadrant angles 0, π/2, π, and 3π/2 ? 9

10. Angles in Other Quadrants  The values of trig functions of angles greater than 90° (or less than 0° ) can be determined from their values at corresponding acute angles called reference angles. 10

11. Definition of Reference Angle  Let θ be an angle in standard position. Its reference angle θ ' is the acute angle formed by the terminal side of θ and the x -axis. 11

12. Example 3  Find the reference angle θ ' for each. a. 300° b. 2.3 rad. c. –135° 12

13. Trig Functions of Any Angle  To find the value of a trig function of any angle θ: 1. Determine the function value for the associated reference angle θ '. 2. Depending on the quadrant in which θ lies, use the appropriate +/- sign for the function value. 13

14. Trig Values of Common Angles 14

15. Study Tip  A pattern for the sine function that may help you remember the values for the common angles.  Reverse the order to get cosine values of the same angles. 15

16. Example 4  Evaluate each trig function. 16

17. Example 5  Let θ be an angle in Quadrant II such that. Find cos θ by using trig identities. 17

18. Homework 4.4  Worksheet 4.4 18