1. Trigonometric Functions of Any Angle
Demana, Waits, Foley, Kennedy
4.3
Trigonometric Functions of Any Angle

2. What you’ll learn about
Trigonometric Functions of Any Angle
Trigonometric Functions of Real Numbers
Periodic Functions
The 16-point unit circle
… and why
Extending trigonometric functions beyond triangle
ratios opens up a new world of applications.

3. Initial Side, Terminal Side

4. Positive Angle, Negative Angle

5. Coterminal Angles
Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

6. Example: Finding Coterminal Angles

7. Solution

8. Example: Finding Coterminal Angles

9. Solution

10. Example: Evaluating Trig Functions Determined by a Point in QI

11. Solution

12. Trigonometric Functions of any Angle

13. Reference Angles Reference Angle
For any given angle, its reference angle is an acute version of that angle. In standard position, the reference angle is the smallest angle between the terminal side and the x-axis. The values of the trig functions of angle θ are the same as the trig values of the reference angle for θ, give or take a minus sign.

14. Evaluating Trig Functions of a Nonquadrantal Angle θ
Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.
Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.
Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.

15. Evaluating Trig Functions of a Nonquadrantal Angle θ
4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.
5. Use the coordinates of point P and the definitions to determine the six trig functions.

16. Example: Evaluating More Trig Functions

17. Solution

18. Example: Using One Trig Ratio to Find the Others

19. Solution

20. Solution Continued

21. Unit Circle
The unit circle is a circle of radius 1 centered at the origin.

22. Trigonometric Functions of Real Numbers

23. Periodic Function

24. The 16-Point Unit Circle