NAME DATE PERIOD 12-6 Study Guide and Intervention 12... · Study Guide and Intervention...
Transcript of NAME DATE PERIOD 12-6 Study Guide and Intervention 12... · Study Guide and Intervention...
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ght
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lencoe/M
cG
raw
-Hill
, a
div
isio
n o
f T
he
McG
raw
-Hill
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mp
an
ies,
Inc.
Less
on
12-
5
NAME DATE PERIOD
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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
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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
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