1. Warm-Up 8/26 Simplify the each radical expression. 1. 272 2. 3 63
= 16∙17 =4 17
= = 1 7
∙ =

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. 4-3a Trigonometric Functions on the Unit Circle

5. (x, y)
𝑦 𝑟 
sin 𝜃 = r y
𝑥 𝑟
cos 𝜃 = θ x
𝑦 𝑥
tan 𝜃 =

6. Trigonometric Functions of Any Angle

7. Example 1: Let (– 4, 3) be a point on the terminal side of and angle in standard position. Find the exact values of the six trigonometric functions of .
𝑟= − = 25
𝑟=5
= 5 3
= 3 5
cs𝑐 𝜃= 𝑟 𝑦
s𝑖𝑛 𝜃= 𝑦 𝑟
=− 4 5
=− 5 4
se𝑐 𝜃= 𝑟 𝑥
cos 𝜃= 𝑥 𝑟
=− 4 3
=− 3 4
𝑐𝑜𝑡 𝜃= 𝑥 𝑦
ta𝑛 𝜃= 𝑦 𝑥

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 )
r = 1

9. Example 2: Find the exact value of each trigonometric function, if defined. If not defined, write undefined. a. cos  b. tan 450 c. cot 7𝜋 2
= 𝑥 𝑟
= −1 1
=−1
P(– 1 , 0), r = 1
= 𝑦 𝑥
= 1 0
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
450 – 360 = 90
P(0 , 1), r = 1
= 𝑥 𝑦
= 0 −1 =0
7𝜋 2 − 4𝜋 2 = 3𝜋 2
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 
3𝜋 4
4𝜋 4 − 3𝜋 4 = 𝜋 4
’= 𝜋 4

12. Quadrant II Quadrant I Quadrant III Quadrant IV sin : cos : tan : +
(cos, sin)
Quadrant II
Quadrant I
(–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

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

16. −1
math!
4-3a Assignment: TX p251, 2-32 even