Warm-Up 3/27-281. Find the measure of .

sin 𝜃=1416

𝜃=sin−1( 1416 )

𝜃=61°

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.

Trig 3: Trigonometric

Functions on the Unit Circle

θ

(x, y)

r y

x

sin 𝜃=¿ 𝑦𝑟

cos𝜃=¿𝑥𝑟

tan𝜃=¿ 𝑦𝑥, 𝑥≠0

𝑟=√𝑥2+𝑦2

Trigonometric Functions of Any Angle

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 .

s 𝑖𝑛𝜃=𝑦𝑟 ¿−610

cos𝜃=𝑥𝑟 ¿

810

ta𝑛𝜃=𝑦𝑥 ¿−68

¿− 35

¿45

¿− 34

𝑟=√82+ (−6 )2=√100𝑟=10

8

– 6 10

’

• 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 )

2

32

0 or 2r = 1

Example 2: Find the exact value of each trigonometric function, if defined. If not defined, write undefined.

a. sin(– 180)

b. tan

P(– 1 , 0), r = 1

¿𝑦𝑟¿

01¿0

P(0 , – 1), r = 1

¿𝑦𝑥 ¿−10 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

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

Example 3: Sketch each angle. Then find the reference angle.

a. – 150

b.

x

y

x

y

= – 150 180 – 150 = 30 ’= 30

𝜃=3𝜋4’=

4𝜋4 − 3𝜋

4 =𝜋4

θ

(x, y)

r y

x

sin 𝜃=¿𝑦𝑟

cos𝜃=¿𝑥𝑟

tan𝜃=¿𝑦𝑥, 𝑥≠0

𝑟=1sin 𝜃=𝑦cos𝜃=𝑥

(cos, sin)𝑟=√𝑥2+ 𝑦2

tan𝜃=sin 𝜃cos𝜃

x

y

Quadrant IQuadrant II

Quadrant III Quadrant IV

(+x, +y)(–x , +y)

(–x , –y) (+x , –y)

sin : cos : tan :

+++

sin : cos : tan :

+– –

sin : cos : tan :

–– +

sin : cos : tan :

–+ –

(cos, sin)

ALLStudents

Take Calculus

tan𝜃=si n 𝜃cos𝜃

2

32

2

EVALUATING TRIG FUNCTIONS AT ANY ANGLE

1.Find the reference angle.2.Find the corresponding trig value.3.Determine the sign of the angle.

x

y

O

’

12√32√33

12

√32

√3

√22√22

1

Example 4: Find the exact value of each expression.

a. cos (– 240)

x

y

’= 60

= – cos 60 ¿−12

Checkpoints:1. Find the exact values of the three basic trig functions of given (– 4 , 3).

r = 5,

2. Find the exact value of the trig function .

r = 1,

3. Sketch the angle and then find the reference angle of 300.

4. Find the exact value of the trig function .

Quadrant III tan is positive.

Assignment4-3 Worksheet 1-19 all

Warm-Up 3/28

1. Find the exact value of the trig function .

r = 1,

2. Find the exact value of the trig function .

Quadrant II cos is negative.

Assignment4-3 Worksheet 1-19 all