Download Chapter 5 The Trigonometric Functions 5

Chapter 5 The Trigonometric Functions
5-1 Angles and Degree Measure
Pages 280–283
45
60
1. If an angle has a positive
measure, the rotation is in a
counterclockwise direction. If an
angle has a negative measure,
the rotation is in a clockwise
direction.
3. 270° + 360k° where k is an
integer
4.
y
O
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
35.
34° 579
2128.513°
2720°
22° 1 360k°; Sample answers:
382°; 2338°
93°; II
47°
15°; 0.25° or 159; about 0.0042°
or 150
168° 219
286° 529 480
246° 529 33.60
214.089°
173.410°
1002.508°
720°
22700°
22070°
Glencoe/McGraw-Hill
26
3600
2. Add 29, } , and } .
14.
16.
18.
1260°
272° 469 300
29.102°
1620°
2170° 1 360k°;
Sample answers: 190°; 2530°
282°; IV
30°
216° 459
20.
22.
24.
26.
28.
30.
32.
34.
36.
2183° 289 120
27° 279 540
23.242°
233.421°
2405.272°
21080°
540°
810°
1440°
6.
8.
10.
12.
122
x
Advanced Mathematical Concepts
Chapter 5
37. 30° 1 360k°;
Sample answers: 390°; 2330°
39. 113° 1 360k°;
Sample answers: 473°; 2247°
41. –199° 1 360k°;
Sample answers: 161°; 2559°
43. 310°
45. 40°; I
47. 220°; III
49. 96°; II
51. III
53. 32°
55. 60°
57. 35°
59. 4500°; 270,000°
61. 17,100°
38. 245° 1 360k°;
Sample answers: 315°; 2405°
40. 217° 1 360k°;
Sample answers: 577°; 2143°
42. –305° 1 360k°;
Sample answers: 55°; 2665°
44. 780°; 21020°
46. 80°; I
48. 339°; IV
50. 91°; II
52. 33°
54. 23°
56. 17°
58. 20°, 160°, 200°, 340°
60. 90k, where k is an integer
62. 1.08 3 107 to 3.6 3 107
degrees
64. 25° 1 120k°, where k is an
integer
66a. about 3.4 revolutions
66b. 8640°
68. 20
63. 22,320°; 1,339,200°;
80,352,000°; 1,928,448,000°
65a. 44° 269 59.640; 68° 159 41.760
65b. 24.559°; 81.760°
67a. Sample answer: f(x) 5
20.0003x 3 1 0.0647x 2 2
3.5319x 1 76.0203
67b. Sample answer: about 32%
69. 0, 24
71. x 3 1 x 2 2 80x 2 300 5 0
73. point discontinuity
70. 25
72. about 4.91
74.
decreasing for x , 21,
increasing for x . 21
Glencoe/McGraw-Hill
123
Advanced Mathematical Concepts
Chapter 5
75. expanded vertically by a factor of
3, translated down 2 units
76.
y
(3, 5)
( 1, 5)
(0, 3)
x
O
(21, 5), (3, 5), (0, 3)
78. D
77. 0.56x
5-2 Trigonometric Ratios in Right Triangles
Pages 287–290
1. The side opposite the acute
angle of a right triangle is the
side that is not part of either
side of the angle. The side
adjacent to the acute angle is
the side of the triangle that is
part of the side of the angle, but
is not the hypotenuse.
2. cosecant; secant; cotangent
a
b
c
c
a
c
tan A 5 } , csc A 5 } ,
b
a
c
b
sec A 5 } , cot A 5 }
b
a
3. sin A 5 } , cos A 5 } ,
514
w
514
514
w
514
4. sin A 5 cos B, csc A 5 sec B,
tan A 5 cot B
Ï
Ï
5. }} ; }} ; }
15
17
15
17
5
2
6. }
3
w
Ï91
8. sin P 5 } , cos P 5 } ,
1
1.5
7. } < 0.6667
10
10
91
w
91
Ï
tan P 5 } , csc P = } ,
3
10
3
10Ï91
w
w
Ï91
sec P 5 } , cot P 5 }
91
9. It = 0.5Io
Glencoe/McGraw-Hill
3
5
4
5
3
3
4
10. } ; } ; }
124
Advanced Mathematical Concepts
Chapter 5
5Ï89
w 8Ï89
w 5
11. } ; } ; }
3 Ï91
w 3Ï91
w
12. } ; } ; }
13. tangent
14. 3
89
8
89
10
7
3
1
17. } 5 0.4
2.5
10
91
9
5
16. }
15. }
1
0.75
18. } < 1.3333
527
w
24
Ï
20. sin R 5 } , cos R 5 } ,
1
0.125
19. } 5 8
7
24
527
w
7
Ï
tan R 5 } ,
527
w
527
Ï
csc R 5 }} , sec R 5 } ,
24
24
7
527
w
527
Ï
cot R 5 }
7
19
w
Ï39
21. sin R 5 } , cos R 5 } ,
20
154
w
44
20
39
w
22
w
44
Ï
Ï
22. sin R 5 } , cos R 5 } ,
9
7
w
154
w
7
Ï
tan R 5 } , csc R 5 } ,
Ï
Ï
tan R 5 } , csc R 5 } ,
20Ï39
w
w
Ï39
sec R 5 } , cot R 5 }
Ï
Ï
sec R 5 } , cot R 5 }
19
39
20
19
39
9
u
72°
74°
76°
78°
80°
82°
84°
86°
88°
sin
0.951
0.961
0.970
0.978
0.985
0.990
0.995
0.998
0.999
19
24a.
24b.
24c.
24d.
26.
cos
0.309
0.276
0.242
0.208
0.174
0.139
0.105
0.070
0.035
7
w
9
9
7
0.7963540136
0.186524036
35.34015106
1.37638192
u
18°
16°
14°
12°
10°
8°
6°
4°
2°
sin
0.309
0.276
0.242
0.208
0.174
0.139
0.105
0.070
0.035
cos
0.951
0.961
0.970
0.978
0.985
0.990
0.995
0.998
0.999
tan
0.325
0.287
0.249
0.213
0.176
0.141
0.105
0.070
0.035
26a. 0
26b. 1
26c. 0
25a. 1
25b. 0
Glencoe/McGraw-Hill
22
w
2
23. 1.3
25.
2
125
Advanced Mathematical Concepts
Chapter 5
10
w
Ï
28. cos R 5 } ,
2
27. about 1.5103
7
10
w
20
7
3
10
w
20
sin u
30. tan u 5 }
cos u
2
Ï
tan R 5 } , csc R 5 } ,
3
10
w
Ï
Ï
sec R 5 } , cot R 5 }
7
29a.
29b.
29c.
29d.
about 5.4 m/s
about 5.9 m/s
about 6.4 m/s
increase
3
32. about 4.31 cm
31a. about 87.5°; about 40.5°
31b. about 49.5°; about 2.5°
31c. neither
33. 88° 229 120
35a. 23 employees
35b. $1076
34. 1; 3 or 1
36. 78
1
2
38. C
37. y 5 2 } x 1 6
5-3 Trigonometric Functions on the Unit Circle
Pages 296–298
1. Terminal side of a 180° angle
in standard position is the
negative x-axis which intersects
the unit circle at (21, 0). Since
1
y
2.
y
O
x
1
0
csc u 5 } , csc 180° 5 } which
is undefined.
As u goes from 0° to 90°, the
y-coordinate increases. As u
goes from 90° to 180°, the
y-coordinate decreases.
Glencoe/McGraw-Hill
126
Advanced Mathematical Concepts
Chapter 5
cos u
sin u
x
y
3. cot u 5 } 5 }
4.
Function
Quadrant
I II III IV
1 1 2 2
1 2 2 1
1 2 1 2
sin a or cos a
cos a or sec a
tan a or cot a
5. 0
6. undefined
1
Ï3w
7. sin 30° 5 } , cos 30° 5 } ,
8. sin 225° 5 2 } ,
2
Ï2w
2
2
Ï2w
3
w
Ï
tan 30° 5 } , csc 30° 5 2,
cos 225° 5 2 } , tan 225° 5 1,
2
3
3
w
Ï
sec 30° 5 } , cot 30° 5
2
3
csc 225° 5 2Ï2
w,
Ï3w
sec 225° 5 2Ïw
2, cot 225° 5 1
2
w
2
w
Ï
Ï
10. sin u 5 } , cos u 5 2 } ,
4
3
5
5
4
5
tan u 5 } , csc u 5 } ,
3
4
3
5
sec u 5 } , cot u 5 }
3
4
9. sin u 5 } , cos u 5 } ,
2
2
tan u 5 21, csc u 5
Ï2w,
sec u 5 2Ïw
2, cot u 5 21
2
w
3
w
2
w
Ï
12. sin u 5 } , tan u 5 2Ï3
w,
Ï
Ï
11. sin u 5 2 } , cos u 5 } ,
2
2
2, sec u 5
csc u 5 2Ïw
Ï2w,
2
3
w
Ï
csc u 5 } , sec u 5 22,
2
3
cot u 5 21
Ï3w
cot u 5 2 }
3
14. 1
13. The distances range from about
24,881 miles to 0 miles.
15. 0
17. 21
19. 21
Glencoe/McGraw-Hill
16. undefined
18. 0
20. Sample answers: 0°, 180°
127
Advanced Mathematical Concepts
Chapter 5
Ï2w
Ï2w
22. sin 45° 5 } , cos 45° 5 } ,
21. undefined
2
2
tan 45° 5 1, csc 45° 5
sec 45° 5
Ïw2,
Ï2w, cot 45° 5 1
Ï2w
1
2
24. sin 315° 5 2 } ,
23. sin 150° 5 } ,
2
2
w
Ïw3
Ï
cos 315° 5 } , tan 315° 5 21,
cos 150° 5 2 } , tan 150° 5
2
2
csc 315° 5 2Ïw
2,
Ïw3
2 } , csc 150° 5 2, sec 150°
3
sec 315° 5
2Ï3
w
3
Ïw2, cot 315° 5 21
5 2 } , cot 150° 5 2Ï3
w
1
2
1
2
26. sin 330° 5 2 } , cos 330° 5
25. sin 210° 5 2 } , cos 210° 5
Ï
Ï
2 } , tan 210° 5 } ,
Ïw3
Ï3w
} , tan 330° 5 2 } ,
2
3
csc 210° 5 22, sec 210° 5
csc 330° 5 22, sec 330° 5
3
w
3
w
2
3
2Ï3
w
3
2 } , cot 210° 5
3
2Ïw
} , cot 330° 5 2Ïw
3
3
Ï3w
3
w
Ï
27. sin 420° 5 } , cos 420° 5 } ,
1
2
2
28. 21
3
w
Ï
tan 420° 5 Ïw
3, csc 420° 5 } ,
2
3
3
w
Ï
sec 420° 5 2, cot 420° 5 }
3
3
5
4
5
30. sin u 5 2 } , cos u 5 2 } ,
29. 2
3
4
5
3
4
5
sec u 5 2 } , cot u 5 }
4
3
tan u 5 } , csc u 5 2 } ,
2
w
2
w
Ï
Ï
31. sin u 5 } , cos u 5 2 } ,
2
tan u 5 21, csc u 5
32. sin u 5 0, cos u 5 1, tan u 5 0,
csc u is undefined, sec u 5 1,
cot u is undefined.
2
Ïw2,
sec u 5 2Ï2
w, cot u 5 21
Glencoe/McGraw-Hill
128
Advanced Mathematical Concepts
Chapter 5
65
w
65
65
w
65
Ï
Ï
33. sin u 5 2 } , cos u 5 } ,
8
w
Ï65
34
w
34
Ï
cos u 5 } , tan u 5 2 } ,
5
tan u 5 28, csc u 5 2 } ,
8
sec u 5
34
3Ïw
34
34. sin u 5 2 } ,
3
5
34
w
3,
34
w
5
Ï
Ï
csc u 5 2 } sec u 5 } ,
1
w, cot u 5 2 }
Ï65
8
5
3
cot u 5 2 }
15
17
8
17
35. sin u 5 } , cos u 5 2 } ,
15
8
36. The sine of one angle is the
negative of the sine of the other
angle.
17
15
tan u 5 2 } , csc u 5 } ,
17
8
8
15
sec u 5 2 } , cot u 5 2 }
5
12
5
13
38. sin u 5 2 } , tan u 5 } ,
37. in Quadrant III or IV
13
5
13
12
csc u 5 2 } , sec u 5 2 } ,
12
5
cot u 5 }
3
w
Ï
39. sin u 5 } , cos u 5 2 } ,
1
2
2
2
Ïw3
6
w
w6
12
Ï
Ï
40. cos u 5 } , tan u 5 2 } ,
5
2Ï3
w
3
6
w
tan u 5 2 } , sec u 5 2 } ,
Ï
csc u 5 25, sec u 5 } ,
cot u 5 2Ï3
w
cot u 5 22Ïw
6
3
2Ï5
w
Ï5w
41. sin u 5 } , cos u 5 } ,
5
5
w
2
12
6
w
3
w
Ï
Ï
42. sin u 5 2 } , cos u 5 } ,
5
Ï
csc u 5 } , sec u 5
5
3
3
Ï6w
Ï5w,
tan u 5 2Ïw
2, csc u 5 2 } ,
2
Ï2w
1
2
cot u 5 }
cot u 5 2 }
2
Ï2w
Ï2w
2
2
3
w
Ï
44. }
43. sin u 5 2 } , cos u 5 2 } ,
3
tan u 5 1, csc u 5 2Ïw
2,
sec u 5 2Ïw
2
45. 0° or 90°
Glencoe/McGraw-Hill
46a. k is an even integer.
46b. k is an odd integer.
129
Advanced Mathematical Concepts
Chapter 5
10
w
10
10
w
10
Ï
Ï
48. sin u 5 } , cos u 5 2 } ,
3
47. u 5 0°
10
w
3
Ï
tan u 5 23, csc u 5 } ,
1
3
sec u 5 2Ïw
10, cot u 5 2 }
5
7
49a. 76 ft
49b. 22 ft
49c. 19 ft
50. }
1
2
49d. } r 1 4
52. 23
54. 212.6
51. 240°; III
53. 1.25, 1
55.
6
x2
f (x)
3
4
1 2 21
2
56. } 3
7
f (x)
416
2
15
10
5 O
2
5x
4
6
57.
1 }43 , 2 }32 , }12 2
58. yes; yes; no
*
1
2
59. absolute value; f(x) 5 2 } 2 x
Glencoe/McGraw-Hill
*
60. C
130
Advanced Mathematical Concepts
Chapter 5
5-4 Applying Trigonometric Functions
Pages 301–304
2. Sample answer: Find a.
1a. cos or sec
1b. tan or cot
1c. sin or csc
3. /DCB; /ABC; the measures
are equal; if parallel lines are cut
by a transversal, the alternate
interior angles are congruent.
5. 52.1
7. 12.4
9.
11.
13.
15.
17.
19.
about 743.2 ft
6.3
9.5
18.4
4.0
6; 10.4; 6; 8.5
21a. about 9.9 m
21b. about 6.7 m
21c. about 48.8 m2
1
6
24. V 5 } s 3 tan a
23. about 1088.8 ft
Glencoe/McGraw-Hill
4. Sample answer: If you know
the angle of elevation of the sun
at noon on a particular day, you
can measure the length of the
shadow of the building at noon
on that day. The height of the
building equals the length of the
shadow times the tangent of
the angle of elevation of the
sun.
6. 41.1
8a. about 8.2 cm
8b. about 11.3 cm
8c. about 46.7 cm2
10. 4.5
12. 21.2
14. 76.9
16. 8.6
18. 32.9
20a. about 13.3 cm
20b. about 15.7 cm
20c. about 78.5 cm
22a. about 2.8 cm
22b. 3.2 cm
22c. 19.2 cm
22d. about 26.6 cm2
131
Advanced Mathematical Concepts
Chapter 5
26a. about 37,106.0 ft
26b. about 37,310.4 ft
25a.
84 ft
60˚
8 ft
25b. about 43.9 ft
25c. about 87.8 ft
27. about 366.8 ft; no
28. Let M represent the point of
intersection of the altitude and
E
wF
w. Since nGEF is isosceles,
the altitude bisects E
wF
w. nEMG
is a right triangle. Therefore,
a
s
sin u 5 } or s sin u 5 a and
a
0.5b
tan u 5 } or 0.5b tan u 5 a.
29. Markisha’s; about 7.2 ft
30. about 131.7 ft
3
w
2Ï53
w 7Ï53
w 2
32. } ; } ; }
Ï
31. sin 120° 5 } , cos 120° 5
2
53
53
7
1
2
2 } , tan 120° 5 2Ï3
w,
3
w
Ï
csc 120° 5 } , sec 120° 5
2
3
Ïw3
22, cot 120° 5 2 }
3
33. 43.260
34.
y
y
O
35. $1.32; $0.92
Glencoe/McGraw-Hill
|x
2|
x
36. E
132
Advanced Mathematical Concepts
Chapter 5
Chapter 5 Mid-Chapter Quiz
Page 304
1. 34° 369 180
2. 320°; IV
5
w
Ï11
3. sin G 5 } , cos G 5 } ,
4. sin u 5 2 } ,
6
5Ï29
w
29
6
11
w
11
29
w
29
Ï
tan G 5 } , csc G 5 } ,
Ï
cos u 5 } , tan u 5 2 } ,
6Ï11
w
w
Ï11
sec G 5 } , cot G 5 }
csc u 5 2 } , sec u 5
5
11
6
5
2
5
2
29
Ïw
5
5
2
29
Ïw
} , cot u 5 2 }
5
2
5. about 1043.2 ft
5-5 Solving Right Triangles
Pages 308–312
1a. linear
1b. angle
3. Sample answer:
2. They are complementary.
4. Marta; they need to find the
1
cos
inverse of the cosine, not } .
5. 60°, 300°
6. 150°, 330°
3
w
Ï
7. }
4
3
8. }
2
9. 35.0°
11. A 5 12°, b 5 192.9, c 5 197.2
13. B 5 58°, a 5 6.9, b 5 11.0
10. 53.1°
12. c 5 23.7, A 5 27.6°, B 5 62.4°
14a. about 31.4°
14b. about 1638.3 ft
14c. about 596.9 ft
16. 120°, 300°
18. 90°, 270°
20. 135°, 315°
15. 90°
17. 30°, 330°
19. 225°, 315°
4
5
21. Sample answers: 30°, 150°,
390°, 510°
Glencoe/McGraw-Hill
22. }
133
Advanced Mathematical Concepts
Chapter 5
2
3
5
2
12
26. }
5
24. }
23. }
25. 1
21
w
5
Ï
27. }
29.
31.
33.
35.
37.
39.
41.
43.
28. 59.0°
34.8°
52.7°
36.5°
about 48.8°, 48.8°, and 82.4°
B 5 55°, a 5 5.6, c 5 9.8
c 5 5.7, A 5 42.1°, B 5 47.9°
B 5 38.5°, b 5 10.6, c 5 17.0
B 5 76°, a 5 2.4, b 5 9.5
45a. Since the sine function is the
side opposite divided by the
hypotenuse, the sine cannot be
greater than 1.
45b. Since the secant function is the
hypotenuse divided by the side
opposite, the secant cannot be
between 1 and 21.
45c. Since cosine function is the
side adjacent divided by the
hypotenuse, the cosine cannot
be less than 21.
47a. about 4.6°
47b. about 2.9°
49. about 13.3°
51. y < 36.5, Z < 19.5°, Y < 130.5°
11
w
15
Ï
53. sin F 5 } , cos F 5 } ,
4
7
15
30. 42.8°
32. 65.1°
34. about 36.9° and 53.1°
36. b 5 21.4, A 5 44.4°, B 5 45.6°
38. A 5 43°, a 5 11.7, c 5 17.1
40. a 5 8.7, A 5 67.1°, B 5 22.9°
42. A 5 57°, a 5 12.7, b 5 8.3
44a. about 39.4°
44b. about 788.5 ft
46. about 14.9°
48. about 1.2°
50. about 21.0°
52. about 3587.2 ft
54. 20.3, 1.4, 4.3
11
w
Ï
tan F 5 } , csc F 5
7
15Ï11
w
},
4
44
15
7Ï11
w
sec F 5 } , cot F 5 }
7
Glencoe/McGraw-Hill
44
134
Advanced Mathematical Concepts
Chapter 5
56. (5, 23), (5, 4), (3, 6), (1, 3),
(2, 22)
58. y 5 20.29x 1 587.7
60. A
55. y-axis
3
4
2 21 0
57. 3 21 1
2 8 25
2
2
59. y 5 2 } x 1 2; 2 } ; 2
5
5
5-6 The Law of Sines
Pages 316–318
2. Sample answer:
1.
x
2x
xÏ3
w
} 0 } 0 }
sin 30°
sin 90°
sin 60°
xÏ3
w
x
2x
}
0 }
1 0 }
Ïw3
1
}}
2
}}
2
2x 5 2x
3. K 5 ab sin X
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
5 2x
C 5 81°, a 5 9.1, b 5 12.1
about 18.7
30.4 units2
B 5 70°, b 5 29.2, c 5 29.2
C 5 120°, a 5 8.8, c 5 18.1
A 5 93.9°, b 5 3.4, c 5 7.2
about 97.8
29.6 units2
5.4 units2
25.0 units2
Glencoe/McGraw-Hill
4. Both; if the measures of two
angles and a non-included side
are known or if the measures of
two angles and the included side
are known, the triangle is
unique.
6. C 5 96.8°, b 5 15.5, c 5 18.6
8. 82.2 units2
10. about 303.7 ft
12. A 5 30°, a 5 19.6, b 5 38.6
14. C 5 65°, a 5 12, b 5 10.1
16. B 5 76°, a 5 13.5, b 5 20.7
18. about 17.9
20. 8.7 units2
22. 13,533.9 units2
24. 181.3 units2
135
Advanced Mathematical Concepts
Chapter 5
25. about 234.8 cm2
27. about 70.7 ft2
26. about 192.6 in2
28a. 45°
28b. about 112.7 ft and 72.7 ft
28c. about 265.4 ft
30. about 213,987.7 ft2
29. Applying the Law of Sines,
m
n
r
} 5 } and } 5
sin M
sin N
sin R
s
m sin N
} . Thus sin M 5 }
sin S
n
r sin S
and sin R 5 } . Since /M
s
> /R, sin M 5 sin R and
m sin N
r sin S
} 5 } . However,
n
s
/N > /S and sin N 5 sin S,
m
n
r
s
m
r
n
s
so } 5 } and } 5 } .
Similar proportions can be
derived for p and t. Therefore,
nMNP > nRST.
31a.
31b.
33a.
33b.
about
about
about
about
3.6 mi
1.4 mi
227.7 mi
224.5 mi
32. about 807.7 ft
34. about 6.7 ft
a
b
sin A
sin B
a
sin A
} 5 }
b
sin B
35a. } 5 }
Glencoe/McGraw-Hill
35b.
136
a
} 5
sin A
a
} 5
c
a
} 215
c
a
c
} 2 } 5
c
c
a2c
} 5
c
c
}
sin C
sin A
}
sin C
sin A
} 21
sin C
sin A
sin C
} 2 }
sin C
sin C
sin A 2 sin C
}}
sin C
Advanced Mathematical Concepts
Chapter 5
35c. From Exercise 34b,
35d.
a2c
sin A 2 sin C
} 5 }} or
c
sin C
sin C
sin A 2 sin C
}} 5 } .
c
a2c
a
c
} 5 }
sin A
sin C
a
sin A
} 5 }
c
sin C
a
sin A
} 115 } 11
c
sin C
a
c
sin A
sin C
} 1 } 5 } 1 }
c
c
sin C
sin C
a1c
sin A 1 sin C
} 5 }}
c
sin C
sin C
sin A 1 sin C
} 5 }}
c
a1c
a
} 5
sin A
a
} 5
b
a
} 115
b
a
b
} 1 } 5
b
b
a1b
} 5
b
b
} 5
a1b
b
}
sin B
sin A
}
sin B
sin A
} 11
sin B
sin A
sin B
} 1 }
sin B
sin B
sin A 1 sin B
}}
sin B
sin B
}}
sin A 1 sin B
Therefore,
sin A 2 sin C
sin A 1 sin C
}} 5 }}
a2c
a1c
a1c
sin A 1 sin C
or } 5 }} .
a2c
sin A 2 sin C
36. about 66.0°
35
w
6
35
w
35
Ï
Ï
37. cos u 5 } , tan u 5 2 } ,
38. 83° 1 360k°
35
w
35
Ï
csc u 5 26, sec u 5 } ,
6
cot u 5 2Ï35
w
39. 4 standard carts, 11 deluxe carts
41.
Glencoe/McGraw-Hill
137
40. (0, 4, 22)
42. A
Advanced Mathematical Concepts
Chapter 5
5-7 The Ambiguous Case for the Law of Sines
Pages 324–326
1. A triangle cannot exist if A , 90°
and a , b sin A or if A $ 90°
and a # b.
2.
A
B
30˚
56.4˚
6
93.6˚
C
10
B
A
3. Step 1: Determine that there is
one solution for the triangle.
Step 2: Use the Law of Sines to
solve for B.
Step 3: Subtract the sum of
120 and B from 180 to find C.
Step 4: Use the Law of Sines to
solve for c.
5. 0
7. none
9. A 5 37.0°, B 5 13.0°, a 5 13.4
30˚
123.6˚ 6
26.4˚
10
C
4. 1
6. A 5 15.4°, B 5 147.6°, b <
20.2
8. B 5 50.3°, C 5 91.7°,
c 5 13.0; B 5 129.7°,
C 5 12.3°, c 5 2.8
10a.
45 ft
70 ft
10˚
10b. about 39.3°
10c. about 46.4 ft
12. 1
14. 1
16. 2
11. 0
13. 0
15. 0
Glencoe/McGraw-Hill
138
Advanced Mathematical Concepts
Chapter 5
18. none
20. B 5 90°, C 5 60°, c 5 6.9
17. 2
19. B 5 71.1°, C 5 50.9°,
c 5 23.8; B 5 108.9°,
C 5 13.1°, c 5 6.9
21. A 5 78.2°, B 5 31.8°, b 5 13.5;
A 5 101.8°, B 5 8.2°, b 5 3.6
23. none
22. C 5 80°, a 5 13.1, b 5 17.6
25. B 5 30.1°, C 5 42.7°, b 5 9.0
27. A 5 27.2°, B 5 105.8°, b 5 21.1
29. none
32. A < 70.9°, B 5 55°, C < 54.1°
34a. a , 7
34b. a 5 7 or a $ 14
34c. 7 , a , 14
36a. about 17.2° east of north
36b. about 6 hr
36c. no
38. about 10.8 cm
40. about 305.2 in2
31. about 63.9 units and 41.0 units
33. about 100.6°
35. about 9.6°
37. about 4.1 min
39a. B . 44.9°
39b. B < 44.9°
39c. B , 44.9°
Glencoe/McGraw-Hill
24. A 5 75.9°, C 5 68.1°,
a 5 31.3; A 5 32.1°,
C 5 111.9°, a 5 17.2
26. none
28. A 5 73.3°, C 5 66.7°,
a 5 62.6; A 5 26.7°,
C 5 113.3°, a 5 29.3
30.
139
Advanced Mathematical Concepts
Chapter 5
1 6 i 47
w
4
41. about 185.6 m
Ï
42. 3; } , }}
43. no;
44. (27, 222)
3x
}} 1 1
x21
}}
3x
3 }}
x21
1
2
5
5
1
2
3x
x21
}} 1 }}
x21
x21
}}
9x
}}
x21
4x 2 1
}}
x21
}
9x
}}
x21
4x 2 1
9x
5 }
46. 7
45. 5x 1 2y 5 222
5-8 The Law of Cosines
Pages 330–332
1. The Law of Cosines is needed
to solve a triangle if the
measures of all three sides or
the measures of two sides and
the included angle are given.
3. If the included angle measures
90°, the equation becomes
c 2 5 a 2 1 b 2 2 2ab cos C.
Since cos 90° 5 0,
c 2 5 a 2 1 b 2 2 2ab(0) or
c 2 5 a 2 1 b 2.
2. Sample answer: 1 in., 2 in.,
4 in.
4. Sample answers:
A
10
c
C
53˚
a
B
C
A 5 37°, a ≈ 7.5, c ≈ 12.5
Glencoe/McGraw-Hill
140
Advanced Mathematical Concepts
Chapter 5
4. (continued)
C 5 80°, b ≈ 8.6, c ≈ 12.0
5.
7.
9.
11.
A ≈ 78.4°, B ≈ 51.6, c ≈ 7.8
A 5 43.5°, B 5 54.8°, C 5 81.7°
about 81.0°
102.3 units2
B 5 44.2°, C 5 84.8°, a 5 7.8
13. A 5 34.1°, B 5 44.4°,
C 5 101.5°
15. A 5 51.8°, B 5 70.9°, C 5 57.3°
17. about 13.8°
19. 11.6 units2
21. 290.5 units2
23. 11,486.3 units2
25a. about 68.1 in.
25b. about 1247.1 in2
27. about 342.3 ft
29a. about 122.8 mi
29b. about 2.8 mi
31. the player 30 ft and 20 ft from
the posts
16. A 5 66.9°, B 5 33.8°, c 5 23.0
18. about 91.7 cm and 44.6 cm
20. 107.8 units2
22. 690.1 units2
24. 66.1 units2
26a. about 211.2 cm2
26b. about 110.2°, 69.8°, 110.2°,
69.8°
28. about 31.6 ft
30. about 46,468.5 ft2
32a. about 191,335.4 ft
32b. about 286,609.8 ft
32c. about 96,060.0 ft
34. about 39.2°
36. 210
33. 2
35. 55°
4
3
37. }
Glencoe/McGraw-Hill
A 5 9.1°, B 5 10.9°, c 5 54.2
6.4 units2
about 46.1 ft
A 5 44.4°, B 5 57.1°,
C 5 78.5°
14. A 5 71.6°, C 5 45.4°, b 5 15.0
6.
8.
10.
12.
38. A
141
Advanced Mathematical Concepts
Chapter 5