10.
540°"
180°
#!
or 270°
! !!
" radians "
2"
1260°"
180°
# $!
or $315°
6. $!7"! radians ! !!
" radians "
4"
4
180°
360° # 114.6°
#!
7. 2 radians ! !!
"!
" radians "
5.
3"
!!
2
radians
sin A
!
a
sin & # !y! # !0! or 0
%2%%
) %
%%
0%
# $($
2
2
%
# $4
182 % 192 $ 152
!!
2(18)(19)
cos A #
8. x # $2, y # 0
%%y2%
r # $x%2%%
cosines; a2 # b2 % c2 $ 2bc cos A
152 # 182 % 192 $ 2(18)(19) cos A
2(18)(19) cos A # 182 % 192 $ 152
r
2
$2
x
cos & # !! # !
! or $1
r
2
0
y
tan & # !! # !! or 0
x
$2
sin 47.7°
!!
15
cos A # 0.6725
A # 47.7°
sin B
#!
b
#
sin B #
#2
9. P # I V cos &
10. A ( 90°, 5 ( 12 sin 40°
# (2)(120) cos 70°
no solution
# 82.1; 82.1 watts
11.
sin B
!
18
18 sin 47.7°
!!
15
47.7° % 62.6° % C # 180°
C # 69.7°
B # 62.6°
c2 # a2 % b2 $ 2ab cos C
4.92 # 6.82 % 8.42 $ 2(6.8)(8.4) cos C
2(6.8)(8.4) cos C # 6.82 % 8.42 $ 4.92
6.82 % 8.42 $ 4.92
cos C # !!!
2(6.8)(8.4)
cos C # 0.81223
C # 35.7°
The smallest angle is about 35.7°.
13-5 Law of Cosines
Pages 802 Check for Understanding
1. Sample answer: if you know the measure of three
sides
2. a2 # 82 % 102 $ 2 ' 8 ' 10 cos 40°
3. a, c
4. You do not know two parts of one ratio.
5. See students’ work.
6. See students’ work.
7. cosines; c2 # a2 % b2 $ 2ab cos C
c2 # 112 % 172 $ 2(11)(17) cos 42°
c2 # 132.06
c # 11.5
sin C
sin B
#
A % 81.6° % 42° # 180°
!
!
c
b
A # 56.4°
sin 42°
sin B
! # !
11.5
sin B #
Pages 802–804 Exercises
sin C
!
c
sin C
!
7
sin C # 7
17
17 sin 42°
!!
11.5
sin C
!
c
sin C
!
8
#
#
sin C #
sin B
!
b
sin 71°
!
16
8 sin 71°
!!
16
C # 28.2°
A % 71° % 28.2° # 180°
A # 80.8°
sin A
!
a
#
sin 80.8°
!!
a
#
a#
6.5
sin A #
sin B
!
b
#
sin 34°
!
12
#
9. sines;
sin C #
sin C
!
c
sin C
!
15
15 sin 34°
!!
12
sin 70°
!
14
C # 44.3°
#
sin B
!
b
sin 34°
!
12
#
sin 101.7°
!!
b
b#
#
a#
34° % B % 44.3° # 180°
B # 101.7°
sin A
!
a
11
11 sin 35°
!!
6.5
A # 76.1
sin C
14. sines;
#
!
c
sin 71°
!
16
16 sin 80.8°
!!
sin 71°
a # 16.7
sin A
!
a
4.5
sin 40°
!
4.5
C # 89.2°
13. cosines;
c2 # a2 % b2 $ 2ab cos C
c2 # 112 % 10.52 $ 2(11)(10.5) cos 35°
c2 # 42.03
C # 6.5
sin C
sin A
#
76.1° % B % 35° # 180°
!
!
c
a
B # 68.9°
sin 35°
sin A
! # !
B # 81.6°
8. sines;
a2 # b2 % c2 $ 2bc cos A
a2 # 62 % 72 $ 2(6)(7) cos 40°
a2 # 20.65
a # 4.5
sin A
#!
40° % B % 89.2° # 180°
a
B # 50.8°
sin 40°
#!
12. cosines;
sin A
!
a
sin 40°
!
a
sin B
!
b
#
sin 70°
!
b
#
14 sin 40°
!!
sin 70°
b#
sin C
!
c
sin 70°
!
14
14 sin 70°
!!
sin 70°
a # 9.6
b # 14
40° % B % 70° # 180°
B # 70°
15. cosines;
a2 # b2 % c2 $ 2bc cos A
1402 # 1852 % 1662 $ 2(185)(166) cos A
2(185)(166) cos A # 1852 % 1662 $ 1402
12 sin 101.7°
!!
sin 34°
cos A #
b # 21.0
1852 % 1662 $ 1402
!!!
2(185)(166)
cos A # 0.6868
A # 46.6°
(Continued next page)
© Glencoe/McGraw-Hill
363
Algebra 2 Chapter 13
sin A
!
a
#
sin B
!
b
sin 46.6°
!!
140
#
sin B
!
185
21. sines; 40° % 59° % C # 180°
C # 81°
sin C
sin A
#
!
!
c
a
sin 81°
!
14
185 sin 46.6°
sin B # !
!
140
B # 73.8°
46.6° % 73.8° % C # 180°
C # 59.6°
sin C
sin A
16. sines;
#!
!
c
a
sin 22°
!
c
#
c#
a#
sin B
!
b
sin 56°
sin 102°
!
!!
b
12.2
12.2 sin 22°
b
!!
sin 56°
#
#
#
sin A
!
a
sin 56°
!
12.2
12.2 sin 102°
!!
sin 56°
61
sin C #
2(12)(5)
sin B
!
12
#
#
sin B #
sin 90°
!
13
12 sin 90°
!!
13
sin B #
sin 78°
!
b
90.0° % 67.4° % C # 180°
C # 22.6°
sin A #
sin 81°
!
14
14 sin 59°
!!
sin 81°
21
61 sin 19°
!!
21
sin 25°
!
13.7
13.7 sin 78°
!!
sin 25°
sin A
!
a
#
sin C
!
c
sin 25°
!
13.7
#
sin 77°
!
c
c#
13.7 sin 77°
!!
sin 25°
2(648)(442)
sin B
!
b
sin B
!
648
cos A # 0.8663
A # 30.0°
sin A
#!
30.0° % 69.2° % C # 180°
a
C # 80.8°
sin 30°
#!
sin B #
345
645 sin 30°
!!
345
B # 69.2°
26. cosines;
a2 # b2 % c2 $ 2bc cos A
21.52 # 16.72 % 10.32 $ 2(16.7)(10.3)
cos A
2(16.7)(10.3) cos A # 16.72 % 10.32 $ 21.52
16.72 % 10.32 $ 21.52
cos A # !!!
2(16.7)(10.3)
20
20 sin 47°
!!
17.9
sin A
!
a
A # 54.8°
20. cosines;
c2 # a2 % b2 $ 2ab cos C
c2 # 21.52 % 132 $ 2(21.5)(13)
cos 38.3°
c2 # 192.56
c # 13.9
sin C
sin A
#
A % 35.4° % 38.3° # 180°
!
!
c
a
A # 106.3°
sin 38.3°
sin B
!! # !
sin B #
b#
sin C
!
c
b # 31.7
c # 31.6
24. 15 % 25 ) 40; no solution
25. cosines;
a2 # b2 % c2 $ 2bc cos A
3452 # 6482 % 4422 $ 2(648)(442) cos A
2(648)(442) cos A # 6482 % 4422 $ 3452
6482 % 4422 $ 3452
cos A # !!!
16
16 sin 35°
!!
10.9
13.9
#
b#
B # 57.3°
19. cosines;
b2 # a2 % c2 $ 2ac cos B
b2 # 202 % 242 $ 2(20)(24) cos 47°
b2 # 321.28
b # 17.9
sin B
sin A
54.8° % 47° % C # 180°
! # !
a
b
C # 78.2°
sin 47°
sin A
! # !
17.9
#
14 sin 40°
!!
sin 81°
b
B # 67.4°
18. cosines;
a2 # b2 % c2 $ 2bc cos A
a2 # 162 % 192 $ 2(16)(19) cos 35°
a2 # 118.96
a # 10.9
sin A
sin B
#
35° % 57.3° % C # 180°
!
!
a
b
C # 87.7°
sin 35°
sin B
! # !
10.9
sin 59°
!
b
C # 71.0°
23. sines; 25° % 78° % C # 180°
C # 77°
sin B
sin A
! # !
a
cos A # 0
A # 90.0°
sin A
!
a
sin 40°
!
a
#
a # 9.1
b # 12.1
22. cosines;
b2 # a2 % c2 $ 2ac cos B
b2 # 512 % 612 $ 2(51)(61) cos 19°
b2 # 438.98
b # 21.0
sin C
sin B
A % 19° % 71.0° # 180°
! # !
c
b
A # 90.0°
sin C
sin 19°
! # !
c # 5.5
b # 14.4
56° % B % 22° # 180°
B # 102°
17. cosines;
a2 # b2 % c2 $ 2bc cos A
132 # 122 % 52 $ 2(12)(5) cos A
2(12)(5) cos A # 122 % 52 $ 132
122 % 52 $ 132
cos A # !!
sin B
!
b
#
sin B
!
b
sin 103°
!!
21.5
cos A # $0.2246
A # 103°
sin B
#!
103° % 49.2° % C # 180°
b
C # 27.8°
sin B
#!
sin B #
16.7
16.7 sin 103°
!!
21.5
B # 49.2°
13
13 sin 38.3°
!!
13.9
B # 35.4°
© Glencoe/McGraw-Hill
364
Algebra 2 Chapter 13
27. cosines;
sin B
!
b
sin B
!
5
32.
a2 # b2 % c2 $ 2bc cos A
a2 # 52 % 4.92 $ 2(5)(4.9) cos 28°
a2 # 5.7
a # 2.4
sin A
#!
28° % 78.0° % C # 180°
a
C # 74.0°
sin 28°
#!
sin B #
85 m
sin B #
cos A #
sin 14.6°
!!
8
46.3˚
242 % 182 $ 82
!!
2(24)(18)
2
# !1!(400)(600) sin 46.3°
2
# 86,756.06; 86,756.06 square feet
a2 # b2 % c2 $ 2bc cos A
a2 # 4002 % 6002 $ 2(400)(600) cos 46.3°
a2 # 188,376.44
a # 434
P # 400 % 600 % 434
# 1434; 1434 feet
36. a2 # b2 % c2 $ 2bc cos A
a2 # 452 % 352 $ 2(45)(35) cos 125°
a2 # 5056.77
a # 71.1; 71.1 miles
37.
200 mi
C # 34.6°
71 cm
106˚
55 cm
74˚
a2
600 ft
A # !1! ab sin C
18
18 sin 14.6°
!!
8
30.
a
400 ft
cos A # 0.9676
A # 14.6°
sin C
#!
14.6° % B % 34.6° # 180°
c
B # 130.8°
sin C
#!
sin C #
702 % 852 $ 502
!!
2(70)(85)
cos A # 0.8088
A # 36°
The smallest angle measures about 36°.
33a. A # 4.08
33b. A # 14.15
33c. A # 9.12
34. Since cos 90° # 1, c2 # a2 % b2 $ 2ab cos A
becomes c2 # a2 % b2.
35.
B # 26.4°
29. cosines;
a2 # b2 % c2 $ 2bc cos A
82 # 242 % 182 $ 2(24)(18) cos A
2(24)(18) cos A # 242 % 182 $ 82
sin A
!
a
C
70 m
a2 # b2 % c2 $ 2bc cos A
502 # 702 % 852 $ 2(70)(85) cos A
2(70)(85) cos A # 702 % 852 $ 502
8.3
7.6 sin 29°
!!
8.3
cos A #
50 m
A
2.4
5 sin 28°
!!
2.4
B # 78.0°
28. cosines;
a2 # b2 % c2 $ 2bc cos A
a2 # 7.62 % 14.12 $ 2(7.6)(14.1) cos 29°
a2 # 69.12
a # 8.3
sin B
sin A
29° % 26.4° % C # 180°
! # !
a
b
C # 124.6°
sin B
sin 29°
! # !
7.6
B
55 cm
71 cm
# b2
% c2 $ 2bc cos A
2
2
a # 55 % 712 $ 2(55)(71) cos 74°
a2 # 5913.27
a # 76.9
a2 # b2 % c2 $ 2bc cos A
a2 # 552 % 712 $ 2(55)(71) cos 106°
a2 # 10218.73
a # 101.1
The diagonals measure 76.9 cm and 101.1 cm
31. q2 # 152 % 152 $ 2(15)(15) cos 123°
q2 # 695.09
q # 26.4; 26.4 centimeters
z 125˚
211 mi
x
238 mi
y
x2 # 2002 % 2382 $ 2(200)(238) cos 125°
x2 # 151248.48
x # 388.9
sin z
!
238
#
sin z #
sin 125°
!!
389
238 sin 125°
!!
389
z # 30°
166° $ 30° # 136°
y2 # 2112 % (388.9)2 $ 2(211)(388.9) cos 136°
y2 # 313819.2
y # 561; 561 miles
© Glencoe/McGraw-Hill
365
Algebra 2 Chapter 13
38.
sin B
!
b
#
sin A
!
a
sin B
!
79
#
sin 45°
!
83
sin B #
Page 804 Mathematics and Society
45° % 42.3° % C # 180°
C # 92.7°
1. Sample answer: measurements of angles,
directions, depths, and distances along each
segment
2. Sample answer: flooding, water leakage, fire,
inadequate ventilation
79 sin 45°
!!
83
B # 42.3°
sin A
!
a
#
sin C
!
c
sin 45°
!
83
#
sin 92.7°
!!
c
c#
13-6 Circular Functions
83 sin 92.7°
!!
sin 45°
Page 808 Check for Understanding
c # 117.3
39. $45°
40.
!
" radians
!!
180°
7!
!!
1! 3! 2! 1!
"
#
# 420
$45"
!
180
1. cos & # x, sin & # y
2. Sample answer: sin & is positive until 180° and is
negative until 360°.
3. Sample answer: Graphs have same shape but
cross the x-axis at different points.
4. See students’ work.
6. sin & # y # $!5!
5. sin & # y # !4!
$"
radians # !
! radians
41.
4
f (x )
y!1
O
f (x ) ! x
x"1
x!1
42. Let h # the height, then w # 2h and ! # 7 % h.
V # !wh
72 # h(2h)(7 % h)
72 # 14h2 % 2h3
0 # 2h3 % 14h2 $ 72
0 # h3 % 7h2 $ 36
h#2
w # 2h # 2(2) or 4
! # 7 % h # 7 % 2 or 9
4 units by 9 units by 2 units
43.
z2 % 4z # 96
2
z % 4z % 4 # 96 % 4
(z % 2)2 # 100
z % 2 # * 10
z # $2 * 10
z # $12 or 8
44. (3 % $2
%)($1
%0
% % $5
%)
# 3$1
%0
% % 3$5
% % $2
%0
% % $1
%0
%
# 3$1
%0
% % 3$5
% % 2$5
% % $1
%0
%
# 4$1
%0
% % 5$5
%
45a. Sample answer: y-intercept: 5000
(1907, 5000)
(1994, 26,689)
y2 $ y1
m#!
!
cos & # x # $!3!
cos & # x # $!12!
sin &
tan & # !
sin &
tan & # !
#
5
cos &
4
!!
5
!
3
$!!
5
#
# $!4!
cos &
5
$!!
13
!
12
$!!
13
12
7. 9
8. cos ($!3"!) # cos $!3"! % 2"
! 4
# cos !!5"!"
4
4
13
# !5!
3
"
$2
%
# $!
!
2
9. sin 660° # sin (300 % 360)°
# sin 300°
$3
%
# $!
!
2
10. yes; 7 white keys
Pages 809–810 Exercises
2
12. sin & # y # $!1!
$2
%
11. sin & # y # !
!
13
2
$2
%
cos & # x # !
!
2
sin &
tan & # !
cos &
$2
%
!!
2
!
# $2%
!!
2
x2 $ x1
#
13
5
x
26,689 $ 5000
!!
1994 $ 1907
# 249
y # mx % b
y # 249x % 5000
45b. Sample answer based on the equation in 45a.
2000 $ 1907 # 93
y # 249x % 5000
# 249(93) % 5000
# 28,157; 28,157 movie screens
cos & # x # !5!
tan & #
#
!
5
!!
13
2
# $!1!
5
#1
13. sin & # y # 0
cos & # x # 1
sin &
tan & # !
#
13
sin &
!
cos &
12
$!!
13
14. 2"
16. 6
cos &
0
!!
1
15. 720°
#0
17. sin 1020° # sin (300 % 720)°
# sin 300°
$3
%
# $!
!
2
© Glencoe/McGraw-Hill
366
Algebra 2 Chapter 13