Copyri

ght

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.

Less

on

12-

5

NAME DATE PERIOD

PDF Pass

Chapter 12 35 Glencoe Algebra 2

Study Guide and InterventionCircular Functions

Circular Functions

The terminal side of angle θ in standard position intersects the unit

circle at P (- 5 −

6 ,

√ ��

11 −

6 ) . Find cos θ and sin θ.

P (- 5 − 6 ,

√ �� 11 −

6 ) = P(cos θ, sin θ), so cos θ = - 5 −

6 and sin θ =

√ �� 11 −

6 .

ExercisesThe terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ.

1. P (- √ � 3

− 2 , 1 −

2 ) 2. P(0, -1)

3. P (- 2 − 3 ,

√ � 5 −

3 ) 4. P (- 4 −

5 , - 3 −

5 )

5. P ( 1 − 6 , -

√ �� 35 −

6 ) 6. P (

√ � 7 −

4 , 3 −

4 )

7. P is on the terminal side of θ = 45°. 8. P is on the terminal side of θ = 120°.

9. P is on the terminal side of θ = 240°. 10. P is on the terminal side of θ = 330°.

x

y

O(-1,0)

(1,0)

(0,-1)

(0,1)

P(cos θ, sin θ)

θ

Defi nition ofSine and Cosine

If the terminal side of an angle θ in standard position

intersects the unit circle at P(x, y), then cos θ = x and

sin θ = y. Therefore, the coordinates of P can be

written as P(cos θ, sin θ).

Example

sin θ = 1 −

2 , cos θ = -

√

�

3 −

2 sin θ = -1, cos θ = 0

sin θ = √

� 5 −

3 , cos θ = -

2 −

3 cos θ = - 4 −

5 , sin θ = - 3 −

5

sin θ = -

√

��

35 −

6 , cos θ = 1 −

6 sin θ =

3 −

4 , cos θ =

√ �

7 −

4

sin θ = √

�

2 −

2 , cos θ =

√

�

2 −

2 sin θ =

√

�

3 −

2 , cos θ = - 1 −

2

sin θ = √

�

3 −

2 , cos θ = -

1 −

2 sin θ = -

1 −

2 , cos θ =

√

�

3 −

2

12-6

021_040_ALG2_A_CRM_C12_CR_660295.indd 35021_040_ALG2_A_CRM_C12_CR_660295.indd 35 12/21/10 10:57 AM12/21/10 10:57 AM

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pa

nie

s, In

c.

NAME DATE PERIOD

PDF Pass

Chapter 12 36 Glencoe Algebra 2

Study Guide and Intervention (continued)

Circular Functions

Periodic FunctionsA periodic function has y-values that repeat at regular intervals. One complete pattern is called a cycle, and the horizontal length of one cycle is called a period.The sine and cosine functions are periodic; each has a period of 360° or 2π radians.

Determine the period of the function.

The pattern of the function repeats every 10 units, so the period of the function is 10.

Find the exact value of each function.

a. sin 855°

sin 855° = sin (135° + 720°)

= sin 135° or √ � 2

− 2

ExercisesDetermine the period of each function.

1. 2.

Find the exact value of each function.

y

O

-1

1

π

θ2π 3π 4π

y

O

-1

1

5 10θ

15 20 25 30 35

Example 1

Example 2

2 4 6 8 1010

y

x

b. cos (

31π

−

6

)

cos ( 31π −

6 ) = cos ( 7π

− 6 + 4π)

= cos 7π −

6 or -

√ � 3 −

2

3. sin (-510°) 4. sin 495° 5. cos (- 5π −

2 )

6. sin ( 5π −

3 ) 7. cos ( 11π

− 4 ) 8. sin (- 3π

− 4 )

5π

−

2

6

√

�

2 −

2

- √

�

2 −

2 -

√

�

3 −

2

0- 1 −

2

- √

�

2 −

2

12-6

021_040_ALG2_A_CRM_C12_CR_660295.indd 36021_040_ALG2_A_CRM_C12_CR_660295.indd 36 12/21/10 10:57 AM12/21/10 10:57 AM