47. cos ! " #2#
3
x " 2, r " 3
r " !x"2"%
""y2"
60. d " 15 & 21
" &6
9 % (&6) " 3
3 % (&6) " &3
&3 % (&6) " &9
&9 % (&6) " &15
!5
"
sin ! " #y# " $#
#
r
3 " !2
"2"%
""y2"
2
9"4%y
5 " y2
$ !5
" "y
48. sec ! " &3
r " 3, x " &1
r " !"x2""
%"y2"
3
61.
107
##
105
" 10 or 100
100 times more acidic
2!"
2
sin ! " #y# " $#
#
r
62.
3
w % 12
##
4w & 16
&
w%4
#
2w & 8
3 " !(&
"1""
)2"
%"y2"
cos ! " #x# " &#1#
r
3
9 " 1 % y2
2
8"y
$ 2!2
""y
49. cos ! " 0
r " 1, x " 0
sin ! " #y# " $#1# or $1
r
1
2
2
r " !"x""
%"y"
1 " !0
"2"%
""y2"
2
1"y
$ 1"y
50. Quadrants I and II
51. Quadrant II
53.
cos !
##
sin !
54a. R
"
x
#r#
#
y
#r#
"#x# '
r
r
#y#
0
H"
"
56.
57.
5)
##
6
52. Quadrant III
y
w % 12
##
4(w & 4)
&
2w % 8
##
4(w & 4)
"
w % 12 & (2w % 8)
###
4(w & 4)
"
&w % 4
##
4(w & 4)
1&5
"#
#
sin 2(45°)
##
32
5
2
y & 1 " &#4#x % #1#
31
%
cos A " #18#
21
sin B " #18#
21
5
S"
a1
##
1&r
"
"
B
6. A " #1# ab sin C
2
" #1#(15)(20)
2
$ %
1
1 & &##
2
4
##
3
#
3
##
2
3
" #8#
9
35°
A
#
" #4# '
8
70°
4
##
3
2
© Glencoe/McGraw-Hill
5
sin B
sin A " #
sin C
1. #
#"#
a#
c#
b
2. Sample answer: two sides and the angle opposite
one of them
3. if A * 90° and a * b sin A or if A + 90° and a * b
4.
5. a + 10
C
31
" &#1#
5
Page 796 Check for Understanding
6)
# 0.581
&2
##
3
#
4
##
3
5
7
y " &#4#x % #1#
13-4 Law of Sines
(25)2 sin2 65°
##
2(32)
)
2
##
2
65. &1.6m % 5" &7.8
&1.6m " &12.8
m"8
v02 sin2 !
##
2g
$
'
y & 1 " &#4#(x & 3)
3 & (&2)
" &#4#
5
a2 % 182 " 212 cos A # 0.8571 sin B # 0.8571
a2 % 324 " 441
A # 31°
B # 59°
a2 " 117
a # 10.8
58. P(boy or red-haired person)
" P(boy) % P(red-haired person) & P(red-haired boy)
" #15# % #5# & #2#
59. r "
"
(100)2
a2 % b2 " c2
"
(w % 4)
##
2(w & 4)
x2 & x1
" #x# " cot !
# 8; 8 feet
900°)
180° radians " #
# or 150°
radians #
#
31
18
##
31
&
63. (x & 6)(x & 4 % 2i)(x & 4 & 2i)
" (x & 6)(x2 & 8x % 20)
" x3 & 8x2 % 20x & 6x2 % 48x & 120
" x3 & 14x2 % 68x & 120
y2 & y1
64. m " #
y & y1 " m(x & x1)
#
# 312.5; 312.5 feet
54b. 45°; 2 ( 45° or 90° yields the greatest value for
sin 2!.
55.
w % 12
##
4(w & 4)
&(w & 4)
##
4(w & 4)
" &#1#
4
tan ! " #y# " $#1# or undefined
x
"
"
v02 sin 2!
"#
g#
"
3, &3, &9, &15
2
sin 63°
# 133.6
A % A % 50° " 180°
2A " 130°
A " 65°
B " 65°
359
2
" #1# (12)(23) sin 100°
2
# 135.9
sin C
#
c
8. A " B
2
##
3
7. A " #1# bc sin A
sin 50°
#c#
"
sin B
#
b
sin 65°
"#
#
c"
12
12 sin 50°
##
sin 65°
c # 10.1
Algebra 2 Chapter 13
sin A
#
a
"
sin B
#
b
sin 40°
#
a
"
sin 80°
#
20
9. 40° % B % 60°
" 180°
B " 80°
a"
sin C
#
c
"
sin B
#
b
sin 60°
#
c
"
sin 80°
#
20
20 sin 40°
##
sin 80°
c"
a # 13.1
sin A
10. 49° % 57° % C #
"
a
" 180°
sin 49°
# "
C " 74°
8
b"
14.
78˚
c
20 sin 60°
##
sin 80°
47° % B % 55° " 180°
B " 78°
B
A
47˚
sin A
#
a
"
sin B
#
b
sin 47°
#
a
"
sin 78°
#
67
a
55˚
67
C
c # 17.6
sin B
#
b
sin A
#
a
sin 49°
#
8
sin 57°
#
b
8 sin 57°
##
sin 49°
"
"
c"
b # 8.9
a"
sin C
#
c
sin 74°
#
c
8 sin 74°
##
sin 49°
sin C
#
c
"
sin 55°
#
c
"
a # 50.1
sin B
#
b
sin 78°
#
67
67 sin 55°
##
sin 78°
c"
c # 10.2
67 sin 47°
##
sin 78°
c # 56.1
B " 78°, a # 50.1, c # 56.1
11. two solutions 8 + 6 + 8 sin 36°
sin B
sin A
# " #
a
b
sin B
#
8
"
sin B "
sin 36°
#
6
8 sin 36°
##
6
Pages 796–798 Exercises
15. A " #1#ab sin C
2
" #1#(18)(13)
2
B # 51.6°
36° % 51.6° % C # 180°
C # 92.4°
sin C
sin A
"
#
#
c
a
sin 92.4°
##
c
sin 36°
##
c#
B # 51.6°, C # 92.4°, c # 10.2
sin C
#
c
"
sin A
#
a
sin 15.6°
##
c
#
sin 36
#
6
c#
C
98˚
64
sin C
#
c
"
sin 98°
#
90
"
sin A "
90
B
44.8° % B % 98° # 180
B # 37.2°
sin B
#
b
sin 37.2°
##
b
"
#
b#
2
" 37.4
21. 49° % 57° % C " 180°
C " 74°
sin A
sin B
"
#
#
a
6 sin 15.6°
##
sin 36°
b
sin 49°
#
8
"
b"
sin 57°
#
b
8 sin 57°
##
sin 49°
b # 8.9
22. 43° % 68° % C " 180°
C " 69°
sin C
sin B
"
#
#
c
sin A
#
a
b
sin A
#
64
64 sin 98°
##
90
sin 69°
#
22
"
b"
A # 44.8°
sin 68°
#
b
22 sin 68°
##
sin 69°
2
" 757.1
20. A " #1#ac sin B
2
" #1#(9.4)(13.5) sin 95°
2
" 63.2
sin A
#
a
"
sin C
#
c
sin 49°
#
8
"
sin 74°
#
c
c"
8 sin 74°
##
sin 49°
c # 10.2
sin C
#
c
"
sin A
#
a
sin 69°
#
22
"
sin 43°
#
a
a"
22 sin 43°
##
sin 69°
b # 21.8
a # 16.1
23. 110° % B % 35.3° #180°
B # 34.7°
sin A
sin C
sin A
sin B
"
"#
#
#
#
a
c
a
sin C
#
c
b
sin 98°
#
90
90 sin 37.2°
##
sin 98°
sin 110°
##
13
sin C
sin C
"#
8
8 sin 110°
" ##
13
C # 35.3°
b # 55.0
A # 44.8°, B # 37.2°, b # 55.0
© Glencoe/McGraw-Hill
2
" #1#(35)(47) sin 67°
" #1#(11.5)(19) sin 20°
c # 2.7
B # 128.4°, C # 15.6°, c # 2.7
12. A , 90°, 6 - 8
no solution
13. c , 98°, 90 + 64
one solution
A
2
" 229.5
19. A " #1# ac sin B
2
c # 10.2
b
2
2
" 52.0
18. A " #1#bc sin A
" #1#(20)(24) sin 73°
36° % 128.4° % C # 180
C # 15.6°
6
6 sin 92.4°
##
sin 36°
2
" #1#(12)(10) sin 120°
sin 34°
" 65.4
17. A " #1# ab sin C
B % 51.6° # 180°
B # 128.4°
16. A " #1#bc sin A
360
sin 110°
##
13
#
b#
sin 34.7°
##
b
13 sin 34.7°
##
sin 110°
b # 7.9
Algebra 2 Chapter 13
30.
24. 36° % 49° % C # 180°
C # 95°
sin B
sin A
# " #
a
sin C
#
c
"
sin 95°
#
c
#
b
sin 49°
#
14.9
"
sin A "
sin A
#
11.6
11.6 sin 49°
##
14.9
c#
A # 36.0°
25. A % 75° % 38° "180°
A " 67°
sin A
sin B
"
#
#
a
sin A
#
a
"
sin C
#
c
"
b"
sin 75°
#
b
sin 67°
#
45
"
sin 38°
#
c
45 sin 75°
##
sin 67°
c"
A
sin B "
14
14 sin 105°
##
18
B # 48.7°
27.
45 sin 38°
##
sin 67°
c#
30˚
sin C
#
c
70˚
b
sin 105°
##
18
18 sin 26.3°
##
sin 105°
sin C
#
c
"
sin B
#
b
sin 30°
"#
a
sin 70°
#
8
"
sin 80°
#
b
a"
8 sin 30°
##
sin 70°
a # 4.3
28.
b"
33.
34.
35.
36.
A
15
sin C
#
c
"
sin 64°
#
17
"
sin B "
64˚
sin B
#
15
15 sin 64°
##
17
sin A
#
a
"
sin 63.5°
##
a
#
a#
sin 64°
#
17
17 sin 63.5°
##
sin 64°
a # 16.9
103° % 31.5° % C # 180°
C # 45.5°
14
103˚
sin 103°
##
14
C
sin B
#
b
sin C
#
c
"
sin B
"#
sin 45.5°
##
c
#
"
sin B "
7.5
7.5 sin 103°
##
14
B # 31.5°
© Glencoe/McGraw-Hill
c#
sin B
#
12
12 sin 45°
##
14
"
sin B
#
b
sin 73°
#
23
"
sin 83°
#
b
b"
23 sin 83°
##
sin 73°
b # 23.9
sin A
#
a
"
sin C
#
c
sin 45°
#
14
"
sin 97.7°
##
c
c#
14 sin 97.7°
##
sin 45°
sin A
#
a
"
sin 25°
#
125
"
sin B
#
b
sin B
#
150
150 sin 25°
##
125
"
sin C
#
c
sin 124.5°
# ##
c
125 sin 124.5°
##
sin 25°
B % 30.5° # 180
B # 149.5°
25° % 149.5° % C # 180°
C # 5.5°
sin A
#
a
"
sin C
#
c
sin 25°
#
125
#
sin 5.5°
##
c
c#
125 sin 5.5°
##
sin 25°
c # 28.4
B # 149.5°, C # 5.5°,
c # 28.4
c # 243.8
B # 30.5°, C # 124.5°,
c " 243.8
37. A * 90°, 10 * 16 sin 40°
no solution
38. A , 90°, 18 - 20
no solution
39. A * 90°, 10 + 8 + 10 sin 40°
two solutions
B % 53.5° # 180°
sin A
sin B
"#
#
a
b
B # 126.5°
sin 40°
sin B
40°
%
126.5°
%
C
# 180°
# " #
8
10
C # 13.5°
10 sin 40°
sin B " ##
sin C
#
c
c
7.5
"
c#
sin B
#
b
23 sin 24°
##
sin 73°
sin A
#
a
B # 37.3°
c # 19.6
A * 90°, 12 * 19 sin 57° no solution
A * 90°, 9 * 20 sin 31° no solution
A , 90°, 12 - 14 no solution
A * 90°, 150 + 125 + 150 sin 25°
two solutions
sin A
#
a
C
B # 52.5°
sin A
#
a
sin 24°
#
c
sin 25°
#
125
29. B
A
"
C
B # 30.5°
25° % 30.5° % C # 180°
C # 124.5°
a
17
sin 73°
#
23
sin B "
8 sin 80°
##
sin 70°
b # 8.4
A % 52.5° % 64° # 180°
A # 63.5°
B
sin C
#
c
sin B "
C
sin A
#
a
"
"
sin 45°
#
14
a
8
sin 70°
#
8
#
sin A
#
a
b
c # 8.3
30° % B % 70° " 180°
B " 80°
B
A
sin 26.3°
##
c
b
24˚
c # 9.8
31. A , 90°, 7 - 9
no solution
32. A * 90°, 14 + 12
one solution
45° % 37.3° % C " 180°
C # 97.7°
sin A
sin B
"#
#
a
b
sin B
"#
73˚
c"
b # 47.2
c # 30.1
26. 105° % 48.7° % C #180°
C # 26.3°
sin A
sin B
sin C
sin A
"
"#
#
#
#
a
c
a
sin 105°
##
18
23
c
c # 19.7
b
sin 67°
#
45
sin B
#
b
sin 49°
#
14.9
14.9 sin 95°
##
sin 49°
73° % B % 24° " 180°
B " 83°
B
sin A
#
a
8
sin 103°
##
14
14 sin 45.5°
##
sin 103°
B # 53.5°
40° % 53.5° % C # 180°
C # 86.5°
c # 10.2
361
(Continued next page)
Algebra 2 Chapter 13
sin A
#
a
"
sin C
#
c
sin 40°
#
8
#
sin 86.5°
##
c
c#
sin A
#
a
"
sin C
#
c
sin 40°
#
8
"
sin 13.5°
##
c
8 sin 86.5°
##
sin 40°
c"
c # 12.4
B
c
36˚
a
22 cm
A
5˚ x
8 sin 13.5°
##
sin 40°
A
B # 126.5°, C # 13.5°,
c # 2.9
A"C
A % 36° % A " 180°
2A " 144°
A " 72°
C
"
sin A
#
a
sin 36°
#
22
"
sin 72°
#
a
a"
sin 58°
#
a
A
"
a"
46.8 miles to Norfolk.
44.
7 cm
6 cm
30˚
C
"
sin 124.2°
##
a
#
46.6˚
66.7˚
sin 26.7
#
y
66.7˚
"
y"
sin 40°
#
7
26.7˚
10
113.3˚
"
sin x° "
sin 40°
#
7
7 sin 26.7
##
sin 40°
sin 40°
#
7
y # 4.9
between 4.9 and 10.4 miles
© Glencoe/McGraw-Hill
"
y"
1
x
&1
y
0
x
&1
y
0
sec 180° " #r# " #1# or &1
&1
cot 180° " #x# " #
# undefined
48. P(6, 6) " 6! or 720
49. Sn " #n#[2a1 % (n & 1)d]
2
0
[2(5) % (50 & 1)25]
S50 " #5#
2
" 25(10 % 1225)
" 30,875
50. A " Pert
" 1000 e0.07(16)
" $3064.85
Yes; Sean has $3064.85 in his account.
sin C
#
c
51.
sin 30°
#
7
6 sin 30°
##
7
3x & 21
#
x2 & 49
.
3x
#
x2 % 7x
2
3x & 21
x % 7x
"#
' #
x2 3(x
& 49
3xx(x % 7)
& 7)
" #1# ' #1#
3x
(x % 7)(x & 7)
"1
2
1
2
x
52. a " 6, c " 2
##
% #y#2 " 1
b2
a
2
2
2
2
b "a &c
x
y2
## % ## " 1
32
36
b2 " 62 & 22
2
b " 36 & 4
b2 " 32
53. 2x2 & 11x & 21 " (2x % 3)(x & 7)
64
54.
" 6(2) & 4(&3)
sin C
#
c
sin 30°
#
7
7 sin 124.2°
##
sin 30°
1
& &3 2 &
" 12 % 12
" 24
sin x°
#
10
10 sin 40°
##
7
x # 66.7°
40˚
r
csc 180° " #r# " #1# undefined
a # 11.5
yes, 11.5 centimeters
1.8 sin 43°
##
sin 90°
x tan 180° " y " #0# " 0
##
(!1, 0) O
B # 25.8°
sin A
#
a
a#
7
"
sin B "
A % 25.8° % 30° # 180°
A # 124.2°
45.
"
#
&1
cos 180° " #x# " #
# or &1
180˚
a # 46.8
sin B
#
6
A
C
sin 43°
#
h
46b. See students’ work.
47.
x " &1, y " 0, r " 1
y
sin 180° " #y# " 0
24.2 sin 58°
##
sin 26°
sin B
#
b
0.25 sin 38°
##
sin 5°
x # 1.8
sin 90°
#
1.8
22 sin 72°
##
sin 36°
B
sin 38°
#
x
h # 1.2 km
sin 26°
#
24.2
a
0.25
x"
h
h#
a # 35.6
41. 8 % 13 * 22 There is no solution.
42. See students’ work.
43.
58° % 96° % C " 180°
C
C " 26°
sin A
sin C
"
#
#
b
a
a
c
58˚ 96˚
24.2 mi B
38˚ 43
˚
"
r
sin B
#
b
P # 35.6 % 35.6 % 22
# 93.2; 93.2 centimeters
sin 5°
#
0.25
B
47˚
c # 2.9
B # 53.5°, C # 86.5°,
c " 12.4
40.
46a.
Page 798 Self Test
sin 73.3°
##
y
1. 49° % B % 90° "180° sin 49° " #7#
tan 41° " #b#
B " 41°
0.8693 # #b#
7 sin 73.3°
##
sin 40
0.7547
c
7
# ##
c
7
7
c # 9.3
b # 6.1
)
radians or ## radians
) radians
$ ##
% " #9108#)0
180°
2
) radians
150)
3. 150° $ ## % " ## radians or #5)# radians
180°
180
6
) radians
135)
4. &135° $ ## % " &## radians or &#3)# radians
180°
180
4
y # 10.4
2. 90°
362
Algebra 2 Chapter 13
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