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Case V 11.27 Solve the triangle ABC, given a
cos A cos A cos A A b2
643.8, b
778.7, and c
cos B cos B cos B B
912.3. See Fig. 11.23.
a2 c2 b2 2ac (643.8)2 (912.3)2 (778.7)2 2(643.8)(912.3)
c2 a2 2bc (778.7)2 (912.3)2 (643.8)2 2(778.7)(912.3) 0.7208 43.88
0.5452 56.96
Fig. 11.23
cos C cos C cos C C Check: A B C 43.88 56.96
b2 c2 2ab (643.8)2 (778.7)2 (912.3)2 2(643.8)(778.7) a2 0.1881 79.16 79.16 180
CHAPTER 11 Oblique Triangles
SUPPLEMENTARY PROBLEMS
11.28 Consider the given parts of triangle ABC. State whether the law of sines or cosines should be used to solve for the required part, and then find its value. (a) a (b) b (c) c (d) A (e) c (f) a (g) a (h) a 17, c 17, a 189, a 0.5, b 315.2, b 25.7, b 7.6, b 14, and B 12, and A 150, and C 0.8, and A 38.7, and C 4.8, and c 30 ; find b. 24 ; find B. 85 18 ; find A. 32.3; find b. 42.45 ; find B. 10.8 ; find c. 7.1; find B. 70 ; find a. Ans. cosines, 8.5 Ans. sines, 35 and 145 Ans. sines, 52 17 Ans. sines, 65.7 Ans. cosines, 0.79 Ans. sines, 78.61 and 101.39 Ans. cosines, 14.3 Ans. cosines, 38
24 18 , B
56 48 , and a 457.8, and A
Solve each of the following oblique triangles ABC, given: 11.29 a 11.30 b 11.31 b 11.32 a 11.33 b 11.34 b 11.35 b 125, A 321, A 215, c 512, b 50.4, c 40.2, a 51.5, a 54 40 , B 75 20 , C 150, B 426, A 33.3, B 31.5, B 62.5, B 65 10 38 30 Ans. b Ans. a Ans. a Ans. c Ans. a Ans. c Ans. c c Ans. b b Ans. a Ans. A Ans. A 139, c 339, c 300, A 680, B 25.1, A 15.7, A 133, C 218, B 60 10 66 10 28 10 92 20 35 30 21 10
42 40 48 50 118 30 112 20 40 40
109 10 , C 38 50 , C 26 0 , C 46 30 , C
78.9, A 52 20 , C 87 0 16.0, A 127 40 , C 11 40 552, B 224, B 344, B 63 10 , B 39 20 , B 85 30 , C 59 10 23 50 , C 120 50 17 50 , C 42 40 , C 46 50 , C 43 30 74 10 93 50
11.36 a
320, c
475, A
35 20
11.37 b 11.38 a 11.39 a
120, c 24.5, b 6.34, b
270, A 18.6, c 7.30, c
118 40 26.4 9.98
11.40 Two ships have radio equipment with a range of 200 km. One is 155 km N42 40 E and the other is 165 km N45 10 W of a shore station. Can the two ships communicate directly Ans. No; they are 222 km apart.
11.41 A ship sails 15.0 mi on a course S40 10 W and then 21.0 mi on a course N28 20 W. Find the distance and direction of the last position from the first. Ans. 20.9 mi, N70 30 W
CHAPTER 11 Oblique Triangles
11.42 A lighthouse is 10 km northwest of a dock. A ship leaves the dock at 9 A.M. and steams west at 12 km/h. At what time will it be 8 km from the lighthouse Ans. 9:17 A.M. and 9:54 A.M.
11.43 Two forces of 115 and 215 lb acting on an object have a resultant of magnitude 275 lb. Find the angle between the directions in which the given forces act. Ans. 70 50
11.44 A tower 150 m high is situated at the top of a hill. At a point 650 m down the hill the angle between the surface of the hill and the line of sight to the top of the tower is 12 30 . Find the inclination of the hill to a horizontal plane. Ans. 7 50
11.45 Three circles of radii 115, 150, and 225 m are tangent to each other externally. Find the angles of the triangle formed by joining the centers of the circles. Ans. 43 10 , 61 20 , 75 30
Solve each of the oblique triangles ABC, given: 11.46 c 11.47 b 11.48 a 11.49 b 11.50 b 11.51 b 78.75, A 730.8, B 31.26, A 13.22, c 10.88, c 86.43, c 33 10 , C 42 13 , C 58 , C 10.00, B 35.73, C 73.46, C 81 25 109 33 Ans. a Ans. a Ans. b Ans. a Ans. a Ans. a a Ans. b b Ans. b Ans. a Ans. A Ans. A 43.57, b 514.5, c 36.47, c 21.47, A 29.66, A 89.52, B 23.19, B 25.40, B 3.56, B 819.2, A 475.9, B 27 28 , B 28 10 , B 72.43, B 1025, A 14.77, B 134 43 , C 48 29 , B 65 25 28 14 98 23 19 20 15 57
23 37 25 57 115 34 49 19
63 10 , A 67 31 116 50 , A 13 51 125 11 , C 30 42 6 35 , C 149 18 23 26 , C 80 24 , C 79 56 , C 34 2 , C 19 2 42 52 72 34 117 48
11.52 a
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