1. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
Section 4.4

2. Objectives: Evaluate trigonometric functions of any angle.
Use reference angles to evaluate trigonometric functions.

3. Unit Circle Rationale What if the radius (hypotenuse) is not 1?
Recall: when using the unit circle to evaluate the value of a trigonometric function, cos θ = x and sin θ = y. Actually, since the radius (hypotenuse) is 1, the trigonometric values are really cos θ = x/1 and
sin θ = y/1.
What if the radius (hypotenuse) is not 1?

4. Exercise 1
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.

5. Exercise 1
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.
The triangle formed by the point
(4, 3) is similar to a smaller triangle in the unit circle.
To get to that unit circle triangle, we need to scale down the larger triangle by dividing by the scale factor 5 (the length of the larger hypotenuse).

6. Exercise 1
Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.
Solution:

7. Trigonometric Functions of Any Angle
Let (x, y) be a point on the terminal side of an angle θ in standard position with

8. Exercise 2
Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trigonometric functions for θ.

9. Signs of Trigonometric Functions
The sign of every trig. function depends on a quadrant (x, y) lies within.
(r is always positive!!!!)

10. Exercise 3 Given sin θ = 4/5 and tan θ < 0, find cos θ and csc θ. 5-3

11. Reference Angles
When building the unit circle, for 120º we drew a triangle with the x-axis to form a 60º angle. This 60º angle was the reference angle for 120º .

12. Reference Angles
Let θ be an angle in standard position. It’s reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis.

13. Reference Angles
Let θ be an angle in standard position. It’s reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis.

14. Exercise 4
Find the reference angle for each of the following:
213°
1.7 rad
−144°

15. Reference Angles
When an angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0º and 360º (or 0 and 2π).

16. Reference Angles How Reference Angles Work:
Same except maybe a difference of sign.

17. Reference Angles
To find the value of a trigonometric function of any angle:
Find the trig value for the associated reference angle.
Pick the correct sign depending on where the terminal side lies.

18. Exercise 5
Evaluate:
sin 5π/3
cos (−60º)
tan 11π/6

19. Exercise 6
Let θ be an angle in Quadrant III such that sin θ = −5/13. Find: a) sec θ and b) tan θ using trigonometric identities.

20. Homework Read Section 4.4, Complete pg. 499-500
# 4-84 (multiples of 4),
# (e), (odd)