1. Precalculus Essentials
Fifth Edition
Chapter 4
Trigonometric Functions
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2. 4.4 Trigonometric Functions of Any Angle

3. Objectives
Use the definitions of trigonometric functions of any angle.
Use the signs of the trigonometric functions.
Find reference angles.
Use reference angles to evaluate trigonometric functions.

4. Definitions of Trigonometric Functions of Any Angle
Let θ be any angle in standard position and let P = (x, y) be a point on the terminal side of θ. If
is the distance
from (0, 0) to (x, y), the six trigonometric functions of θ are defined by the following ratios:

5. Example 1: Evaluating Trigonometric Functions (1 of 3)
Let P = (1, −3) be a point on the terminal side of θ Find each of the six trigonometric functions of θ. Solution: P = (1, −3) is a point on the terminal side of θ.
x = 1 and y = −3

6. Example 1: Evaluating Trigonometric Functions (2 of 3)
Let P = (1, −3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.
Solution:
We have found that

7. Example 1: Evaluating Trigonometric Functions (3 of 3)
Let P = (1, −3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.
Solution:

8. Example 2: Trigonometric Functions of Quadrantal Angles (1 of 4)
Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:
Solution: If θ = 0° = 0 radians, then the terminal side of the angle is on the positive x-axis. Let us select the point P = (1, 0) with x = 1 and y = 0.

9. Example 2: Trigonometric Functions of Quadrantal Angles (2 of 4)
Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:
Solution: If
radians, then the terminal side of the
angle is on the positive y-axis. Let us select the point P = (0, 1) with x = 0 and y = 1.

10. Example 2: Trigonometric Functions of Quadrantal Angles (3 of 4)
Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:
Solution: If
radians, then the terminal side of
angle is on the positive x-axis. Let us select the point P = (−1, 0) with x = −1 and y = 0.

11. Example 2: Trigonometric Functions of Quadrantal Angles (4 of 4)
Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:
Solution: If
radians, then the terminal side of
the angle is on the negative y-axis. Let us select the point P = (0, −1) with x = 0 and y = −1

12. The Signs of the Trigonometric Functions

13. Example 3: Finding the Quadrant in Which an Angle Lies
If sin θ < 0 and cos θ < 0, name the quadrant in which the angle θ lies.
θ lies in Quadrant III.

14. Example 4: Evaluating Trigonometric Functions
Given
find sin θ and sec θ.
Solution: Because both the tangent and the cosine are negative, θ lies in Quadrant II.

15. Definition of a Reference Angle
Let θ be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle
formed by the terminal side of θ and the x-axis.

16. Example 5: Finding Reference Angles
Find the reference angle,
for each of the following angles: a. b. c. d.

17. Finding Reference Angles for Angles Greater Than 360° (2pi) or Less Than −360° (−2pi)
Find a positive angle
less than 360° or
that is coterminal with the given angle.
Draw
in standard position.
Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of
and the x-axis is the reference angle.

18. Example 6: Finding Reference Angles
Find the reference angle for each of the following angles: a. b. c.

19. Using Reference Angles to Evaluate Trigonometric Functions
The values of the trigonometric functions of a given angle, θ, are the same as the values of the trigonometric functions of the reference angle, θ′, except possibly for the sign. A function value of the acute reference angle, θ′, is always positive. However, the same function value for θ may be positive or negative.

20. A Procedure for Using Reference Angles to Evaluate Trigonometric Functions
The value of a trigonometric function of any angle u is found as follows:
Find the associated reference angle, θ, and the function value for θ′.
Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.

21. Example 7: Using Reference Angles to Evaluate Trigonometric Functions (1 of 3)
Use reference angles to find the exact value of sin 300°.
Solution:
Step 1 Find the reference angle,
Step 2 Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.

22. Example 7: Using Reference Angles to Evaluate Trigonometric Functions (2 of 3)
Use reference angles to find the exact value of
Solution:
Step 1 Find the reference angle,
Step 2 Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.

23. Example 7: Using Reference Angles to Evaluate Trigonometric Functions (3 of 3)
Use reference angles to find the exact value of
Solution:
Step 1 Find the reference angle,
Step 2 Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.