1. Chapter 5 Trigonometric Functions
Section 5.3
Trigonometric Functions of Any Angle

2. Trigonometric Functions of Any Angle
Consider angle q in the figure below, which is in standard position with a point P(x, y) on the terminal side of the angle. We define the trigonometric functions based upon this figure.

3. Trigonometric Functions of Any Angle

4. Trigonometric Functions of Any Angle
The value of a trigonometric function is independent of the point chosen on the terminal side of the angle. Consider the figure below. The right triangles formed are similar triangles so the ratios of corresponding sides are equal.

5. Trigonometric Functions of Any Angle
Any point in the coordinate plane can determine an angle in standard position. Consider the figure below:

6. Example 1
Find the value of each of the six trigonometric functions of an angle q in standard position whose terminal side contains the point P(-3, -2).

7. Quadrantal Angles
Recall that a quadrantal angle is an angle who terminal side coincides with the x- or y-axis. Identify the point and then apply the appropriate trigonometric definition to find the value of the quadrantal function.

8. Quadrantal Angles

9. Signs of Trigonometric Functions
The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies.

10. Example 2
Given that tan q = , and sin q < 0, find cos q and csc q.

11. The Reference Angle
Given angle q in standard position, its reference angle q’ is the smallest positive angle formed by the terminal side of angle q and the x-axis.

12. Example 3
For each of the following, sketch the given angle q (in standard position) and its reference angle q’. Then determine the measure of q’.
q = 1200
q = 3450
q = 9240
q = p
q = -4
q = 17

13. Reference Angle Theorem
To evaluate the sin q, determine sin q’. Then use either sin q’ or its opposite as the answer, depending on which has the correct sign.

14. Example 4
Evaluate each function.
sin 2100
cos 4050
tan 3000

15. Assignment
Section 5.3 Worksheet