1. Trigonometric Functions of Any Angle
Section 9.3

2. What You Will Learn Evaluate trigonometric functions of any angle.
Find and use reference angles to evaluate trigonometric functions

3. Take Note

4. Example: Evaluating Trigonometric Functions Given a Point
Let (−4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ.

5. Take Note
It is convenient to use the unit circle to find trigonometric functions of quadrantal angles. A quadrantal angle is an angle in standard position whose terminal side lies on an axis. The measure of a quadrantal angle is always a multiple of 90º, or π/2 radians.

6. Using the Unit Circle Use the unit circle to evaluate the six trigonometric functions of θ = 270º.
Find the values of the six trigonometric functions. Let x = 0 and y = −1 to evaluate the trigonometric functions.

8. Take Note

9. Reference angle
Let θ be an angle in standard position. Its reference angle is the acute angle θ‘ formed by the terminal side of θ and the horizontal axis.

10. Take Note

11. Reference angles are always measured between the x axis and the terminal side of the angle (always +ves !)
Notice the butterfly shape

12. Finding the Trig functions of any angle
Find the reference angle
Determine the value of the given trig function of the reference angle.
Determine the sign—based on the quadrant of the given (original) angle

13. Drawing Reference Angles
Find the reference angle θ', and sketch θ and θ' in standard position.
1) θ = -145° 2)

14. Using Reference Angles to Evaluate Functions
Evaluate (a) tan(−240º) and (b) csc 17π/6 .

15. Now: Find all 6 trig functions of an angle
Find the six trigonometric functions of θ with the given constraint.

16. Now, let’s make it tougher
The terminal side of θ lies on the line y = (1/3)x in quadrant I. Find the values of the 6 trig functions of θ by finding a point on the line.
sin θ =
cos θ =
csc θ =
sec θ =
tan θ =
cot θ =

17. Try These:
Find the values of the six trig functions of θ with the given constraints.
sin θ = 0; where sec θ = -1
Find the reference angle θ‘ and sketch it and the angle θ in standard position a) θ = b) θ = 750°