NAME
DATE
12-6
PERIOD
Study Guide and Intervention
Circular Functions
Circular Functions
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).
Definition of
Sine and Cosine
(0,1) y
P(cos θ, sin θ)
θ
(-1,0)
(1,0)
x
O
(0,-1)
Example
The terminal side of angle θ in standard position intersects the unit
)
(
P(- − , −) = P(cos θ, sin θ), so cos θ = - − and sin θ = −.
√
11
6
6
5
circle at P - −
, − . Find cos θ and sin θ.
5
6
√
11
6
√
11
6
5
6
Exercises
The terminal side of angle θ in standard position intersects the unit circle at each
point P. Find cos θ and sin θ.
√3
2 2
)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. P(0, -1)
√3
1
sin θ = −
, cos θ = - −
2
(
√
5
3
2
3. P - −
,−
3
sin θ = -1, cos θ = 0
2
)
(
√5
3
(6
6
5
( 47 4 )
√
3
6. P −, −
√
35
1
sin θ = - −, cos θ = −
6
√7
3
sin θ = −
, cos θ = −
6
7. P is on the terminal side of θ = 45°.
4
9. P is on the terminal side of θ = 240°.
4
8. P is on the terminal side of θ = 120°.
√3
2
√2
√
2
sin θ = −, cos θ = −
2
2
1
sin θ = −, cos θ = - −
2
10. P is on the terminal side of θ = 330°.
√3
1
sin θ = −, cos θ = - −
2
2
Chapter 12
5
5
3
)
5
3
4
cos θ = - −
, sin θ = - −
2
sin θ = −, cos θ = - −
√
35
1
5. P −
, -−
)
3
4
4. P - −
, -−
√3
2
1
sin θ = - −
, cos θ = −
2
35
Glencoe Algebra 2
Lesson 12-5
(
1
1. P - − , −
NAME
DATE
12-6
PERIOD
Study Guide and Intervention
(continued)
Circular Functions
Periodic Functions
A 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.
Example 1
Determine the period of the function.
The pattern of the function repeats every 10 units,
so the period of the function is 10.
y
1
O
5
-1
15
10
Example 2
20
25
30
35
θ
Find the exact value of each function.
(6)
31π
7π
= cos (−
+ 4π)
cos (−
6 )
6
31π
b. cos −
a. sin 855°
sin 855° = sin (135° + 720°)
√
2
2
= sin 135° or −
√3
7π
or - −
= cos −
6
2
Determine the period of each function.
1.
1
y
O
2.
2π
π
3π
4π
θ
5π
−
y
0
2
2
-1
6
4
8
10
x
6
Find the exact value of each function.
1
2
3. sin (-510°) - −
(3)
5π
6. sin −
Chapter 12
√3
2
-−
√2
2
(4)
11π
7. cos −
√2
2
- −
36
)
(
2
(
4 )
5π
5. cos - −
4. sin 495° −
3π
8. sin - −
0
√2
2
- −
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises