Download Reteach 10.3

Name ________________________________________ Date __________________ Class__________________
LESSON
10-3
Reteach
The Unit Circle
Radians are a real number measure of rotation.
To convert between radians and degrees, use the following identity.
π radians = 180°
To convert from radians to degrees,
solve the identity for 1 radian.
180°
1 radian =
π radians
To convert from degrees to radians,
solve the identity for 1 degree.
π radians
1 degree =
180°
Convert 60° to radians.
⎛ π radians ⎞ π
⎟ = radians
60° = 60° ⎜
⎜ 180° 3 ⎟ 3
⎝
⎠
Convert
Use dimensional analysis to help. Notice
that the degrees cancel so the remaining
unit is radians.
5π
radians to degrees.
4
⎛ 5π
5π
radians = ⎜
4
⎝ 4
⎞ ⎛ 45 180° ⎞
radians ⎟ ⎜
⎟ = 225°
⎠ ⎝ π radians ⎠
The radians cancel so
the remaining unit is
degrees.
Convert each measure from degrees to radians.
1. −45°
2. 150°
⎛ π radians ⎞
−45° = −45° ⎜
⎟
⎝ 180° ⎠
⎛ π radians ⎞
150° = 150° ⎜
⎟
⎝ 180° ⎠
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_________________________________
4. −120°
3. 210°
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_________________________________
Convert each measure from radians to degrees.
5.
4π
radians
3
6. −
4π
⎛ 4π
⎞ ⎛ 180° ⎞
radians = ⎜
radians ⎟ ⎜
⎟
3
⎝ 3
⎠ ⎝ π radians ⎠
−
_____________________________________
7.
π
6
radians
8.
_____________________________________
3π
radians
2
3π
⎛ 3π
⎞ ⎛ 180° ⎞
radians = ⎜ −
radians ⎟ ⎜
⎟
2
⎝ 2
⎠ ⎝ π radians ⎠
_________________________________
5π
radians
3
_________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
10-22
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
10-3
Reteach
The Unit Circle (continued)
Use a reference angle to find the exact value of the sine, cosine, and
tangent of an angle in any quadrant.
Find the value of the trigonometric functions of 150°.
Step 1 Sketch the angle.
Find the measure of the reference angle.
180° − 150° = 30°
Step 2 Draw the triangle with the reference angle.
Label the sides with their lengths.
Step 3 Find the sine, cosine, and tangent of 30°.
1
sin 30° =
2
cos 30° =
tan 30° =
3
2
1
3
The diagram
shows the
quadrants in
which each
trigonometric
function is
positive.
Step 4 Adjust the signs for 150°.
sin 150° =
1
2
cos 150° = −
tan 150° = −
3
2
1
3
Complete to find the exact value of the sine, cosine, and tangent
of 315°.
9. Find the measure of the reference angle.
___________________________________
10. Find the sine, cosine, and tangent of the
reference angle.
_______________________________
_______________________________
_______________________________
11. Adjust the signs to find the sine, cosine, and
tangent of 315°.
_______________________________
_______________________________
_______________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
10-23
Holt McDougal Algebra 2
15.
3
16.
17.
2
2
;−
; −1
2
2
18.
1 3
3
;−
19. − ;
2 2
3
2
2
;−
;1
2
2
21. −
7. 30°
1
3
3
;−
;−
2
2
3
9. 45°
3 1
; ; 3
2 2
20. −
2
2
2
cos 45° =
2
10. sin 45° =
3 1
;− ; 3
2
2
tan 45° = 1
22. 2073 mi
2
2
2
cos 315° =
2
tan 315° = −1
11. sin 315° = −
Practice C
5π
radians
2
1. −270°
2.
3. 50°
4. −
5. 315°
6. −330°
35π
7.
radians
18
8. 63°
9.
π
radians
15
10π
radians
9
12. −
7π
radians
12
1 3
3
13. − ;
;−
2 2
3
14. −
2 2
;
; −1
2 2
15. −
17. −
19.
3 1
;− ; 3
2
2
16.
2
2
;−
;1
2
2
18.
2 2
;
;1
2 2
1
3
3
;−
21. − ; −
2
2
3
23. −
3 1
; ;− 3
2 2
Challenge
1. 6080 ft
2. 1,600,921 mi; 66,705 mi/h
3. Area of circle = πr 2; A sector whose central
angle has a measure of θ radians has an
θ
times the area of the circle. So
2π
θ
1
Area of sector =
πr 2 = θr 2 .
2π
2
10. 234°
37π
radians
30
11.
area of
( )
4.
3 1
;− ;− 3
2
2
b. θ =
2 2
;
; −1
2 2
3.
4
radians
7π
radians
6
5. 240°
2.
2π
π
or
6
3
π π
⋅
2 3
=
π2
6
e. Yes; possible answer: because the arc
length of the fragment is very close to
the arc length that would be expected
for a plate of diameter π
2
2
;−
; −1
2
2
2.
5π
radians
6
4. −
2
d. 1.64 in.
Reteach
π
π
c. S = r θ =
25. 138 ft
1. −
4
1. a. r =
20. 0; −1; 0
24.
π
Problem Solving
1
3
3
;−
;−
2
2
3
22. −
8. 300°
2π
radians
3
1
4
3. C
4. H
5. B
6. F
Reading Strategy
6. −270°
1. 2π
2. 0
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A35
Holt McDougal Algebra 2