1. Warm-Up 3/27-28 1. Find the measure of 𝜃. sin 𝜃= 14 16 𝜃= sin −1 14 16
𝜃=61°

3. Rigor: You will learn how to find reference angles, to evaluate and determine sign of trig functions of any angle. Relevance: You will be able to solve real world problems using reference angles.
MA.912. A.2.11

4. Trig 3: Trigonometric Functions on the Unit Circle

5. 𝑟= 𝑥 2 + 𝑦 2
𝑦 𝑟
(x, y)
sin 𝜃 =  r
𝑥 𝑟 y
cos 𝜃 = θ
𝑦 𝑥
tan 𝜃 =
,𝑥≠0 x

6. Trigonometric Functions of Any Angle

7. Example 1: Let (8, –6) be a point on the terminal side of and angle in standard position. Find the exact values of the three basic trig functions of .
𝑟= −6 2 = 100
𝑟=10 8
= −6 10
=− 3 5
s𝑖𝑛 𝜃= 𝑦 𝑟 ’
– 6 10
= 8 10
= 4 5
cos 𝜃= 𝑥 𝑟
= −6 8
ta𝑛 𝜃= 𝑦 𝑥
=− 3 4

8. r = 1 For 0° or 360°, use the coordinate (1, 0)
For 90°, use the coordinate (0, 1)
For 180°, use the coordinate (– 1, 0)
For 270°, use the coordinate (0, – 1 )

9. Example 2: Find the exact value of each trigonometric function, if defined. If not defined, write undefined. a. sin(– 180) b. tan 3𝜋 2
= 𝑦 𝑟
= 0 1 =0
P(– 1 , 0), r = 1
= 𝑦 𝑥
= −1 0
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
P(0 , – 1), r = 1

10. Reference Angle: an acute angle formed by the terminal side and the x-axis.

11. Example 3: Sketch each angle. Then find the reference angle. a
Example 3: Sketch each angle. Then find the reference angle. a. – 150 b. 3𝜋 4
180 – 150 = 30
’= 30
 = – 150 
𝜃= 3𝜋 4
4𝜋 4 − 3𝜋 4 = 𝜋 4
’= 𝜋 4

12. 𝑟= 𝑥 2 + 𝑦 2 (cos, sin) 𝑦 𝑟 sin 𝜃 = 𝑥 𝑟 cos 𝜃 = 𝑦 𝑥 tan 𝜃 = ,𝑥≠0 𝑟=1
𝑟= 𝑥 2 + 𝑦 2
(cos, sin)
𝑦 𝑟
(x, y)
sin 𝜃 = 
𝑥 𝑟
cos 𝜃 = r y
𝑦 𝑥
tan 𝜃 =
,𝑥≠0 θ
𝑟=1 x
sin 𝜃 =𝑦
cos 𝜃 =𝑥
tan 𝜃 = sin 𝜃 cos 𝜃

13. (cos, sin) tan 𝜃 = sin 𝜃 cos 𝜃 Quadrant II Quadrant I Quadrant III
(–x , +y)
(+x, +y)
sin :
cos :
tan : + –
sin :
cos :
tan : +
tan 𝜃 = sin 𝜃 cos 𝜃
Students
ALL
Quadrant III
Quadrant IV
(–x , –y)
(+x , –y)
sin :
cos :
tan : – +
sin :
cos :
tan : – +
Take
Calculus

14. EVALUATING TRIG FUNCTIONS AT ANY ANGLE Find the reference angle.
Find the corresponding trig value.
Determine the sign of the angle. y  O x ’

15. 2 2
1 2
3 2
3 2
2 2
1 2
3 3 1 3

16. Example 4: Find the exact value of each expression. a. cos (– 240)
=− 1 2
= – cos 60 y
’= 60 x

17. Checkpoints:
1. Find the exact values of the three basic trig functions of  given (– 4 , 3).
r = 5, sin 𝜃 = 3 5 , cos 𝜃 =− 4 5 , tan 𝜃 =− 3 4
2. Find the exact value of the trig function cos 𝜋 .
r = 1, cos 𝜋 = 𝑥 𝑟 = −1 1 =−1
3. Sketch the angle and then find the reference angle of 300.
4. Find the exact value of the trig function tan 7𝜋 6 .
Quadrant III tan  is positive.
tan 𝜋 6 = =

18. Assignment
4-3 Worksheet 1-19 all

19. Warm-Up 3/28 1. Find the exact value of the trig function sin 𝜋 2 .
r = 1, sin 𝜋 = 𝑦 𝑟 = 1 1 =1
2. Find the exact value of the trig function cos 120° .
Quadrant II cos  is negative.
− cos 60° =− 1 2

20. Assignment
4-3 Worksheet 1-19 all