Law of Sines and Cosines- Practice for Wednesday’s Quiz
1. By applying the Law of Sines and the Law of Cosines, try to figure out the total area of the figure below.
A mountain climber is in the base of two mountains. He is trying to decide how much rope is needed to get to the top of the tallest mountain. He knows that from the top of the smaller mountain to the top of the taller mountain that he wants to climb is 100 meters apart and that the smaller mountain is 85 meters tall. He also knows that there is a 37 angle between both mountains. Figure out how much rope is needed to get to the top.
- A farmer owns a plot of land, but has decided that he wants to move to a different town. He wants to sell his land, but part of the paper that had the size of his land on it had been burned up. The piece that remained had the following information: Side a = 300 meters, Side b = 400meters Side c = 600 meters, and Side d = 500 meters. The angle between side b and side c was 76. The rest of the information was missing. Using this information and the diagram below, help the farmer find the total area of his farmland.
- A family decides to go on a vacation and see the country. They start from their home and drive 200 miles in a straight direction until they come to a place where they would like to stay for a while. After they leave this place, they drive 100 more miles at an angle of 115, they again stay at another place. They finally decide they want to come home, but wonder if there is a shorter route, is there? Help them find a way home.
- There is a shipwreck at sea and the ship sends out a signal for rescue to two separate stations. Station A is located 80 miles from station B, and the distress signal was picked up by station A at 43 from the ship, and by station B at 61 from the ship. Find out how far each station is from the ship and which one should send a rescue ship.
Law of Sines and Cosines.
Appendix.
Answers for Practice Questions 1-10.
1. a = b
SinA SinB
27 = 35
SinA Sin48
35SinA = 27Sin48
SinA = 27Sin48
35
A = Sin-1(0.5733)
A = 35
C = 180 – 48 – 35 = 97
c2 = a2 + b2 – 2ab CosC
c2 = 27 + 35 – 2 (27)(35) Cos97
c2 = 729 + 1225 – 2 (945) Cos97
c2 = 1954 – 1890 Cos97
c2 = 1954 +230.33
c2 = 2184.33
c = 46.7
2. a2 = b2 + c2 – 2bc CosA
32 = 52 + 72 – 2 (5)(7) CosA
9 = 25 + 49 – 2 (35) CosA
9 = 74 – 70 CosA
-65 = -70 CosA
0.9286 = CosA
Cos-1 0.9286 = A
A = 22
3.c2 = a2 + b2 – 2ab CosC
c2 = 112 + 152 – 2 (11)(15) Cos88
c2 = 121 + 225 – 2 (165) Cos88
c2 = 346 – 330 Cos88
c2 = 346 + 11.517
c2 = 357.517
c = 18.9
4. a = b
SinA SinB
45 = b
Sin48 Sin62
bSin48 = 45Sin62
b = 45Sin62
Sin48
b = 53.5
C = 180 – 48 – 62 = 70
c2 = a2 + b2 – 2ab CosC
c2 = 452 + 53.52 – 2 (45)(53.5) Cos70
c2 = 2025 + 2862.25 – 2 (2407.5) Cos70
c2 = 4887.25 – 4815 Cos70
c2 = 4887.25 + 1646.827
c2 = 6534.077
c = 80.8
5. b = c
SinB SinC
6.5 = 8.5
Sin41 SinC
6.5SinC = 8.5Sin41
SinC = 8.5Sin41
6.5
C = Sin-1 (0.8579)
C = 59
A = 180 – 41 – 59 = 80
a2 = b2 + c2 – 2bc CosA
a2 = 6.52 + 8.52 – 2 (6.5)(8.5) Cos80
a2 = 42.25 + 72.25 – 2 (55.25) Cos80
a2 = 114.5 – 110.5 Cos80
a2 = 114.5 - 19.188
a2 = 95.312
a = 9.8
6. A = 180 – 41 – 51 = 88
a = c
SinA SinC
a = 100
Sin88 Sin51
aSin51 = 100Sin88
a = 100Sin88
Sin51
a = 128.6
7. b = c
SinB SinC
b = 30
Sin110 Sin45
bSin45 = 30Sin110
b = 30Sin110
Sin45
b = 40
8. a = b
SinA SinB
8 = 5
Sin35 SinB
8SinB = 5Sin35
SinB = 5Sin35
8
B = Sin-1 0.3585
B = 21
C = 180 – 35 – 21 = 124
c2 = a2 + b2 – 2ab CosC
c2 = 8 + 5 – 2 (8)(5) Cos124
c2 = 64 + 25 – 2 (40) Cos124
c2 = 89 – 80 Cos124
c2 = 89 + 44.74
c2 = 133.74
c = 11.6
9.b2 = a2 + c2 – 2ab CosB
b2 = 22.1 + 17.5 – 2 (22.1)(17.5) Cos109
b2 = 488.41 + 306.25 – 2 (386.75) Cos109
b2 = 794.66 – 773.5 Cos109
b2 = 794.66 + 251.83
b2 = 1046.49
b = 32.3
a = b
SinA SinB
22.1 = 32.3
SinA Sin109
32.3SinA = 22.1Sin109
SinA = 22.1Sin109
32.3
A = Sin-10.6469
A = 40
C = 180 – 40 – 109 = 31
10. c2 = a2 + b2 – 2ab CosC
c2 = 302 + 522 – 2 (30)(52) Cos65
c2 = 900 + 2704 – 2 (1560) Cos65
c2 = 3604 – 3120 Cos65
c2 = 3604 – 1318.57
c2 = 4922.57
c = 70.2
c2 = a2 + b2 – 2ab CosC
70.22 = 382 + 422 – 2 (38)(42) CosC
4928.04 = 1444 + 1764 – 2 (1596) CosC
4928.04 = 3208 - 3192 CosC
1720.04 = – 3192 CosC
-0.5389 = CosC
Cos-1 – 0.5389 = C
C = 123
Area 1 = ½ ab SinC
Area 1 = ½ (30)(52) Sin65
Area 1 = ½ 1560 Sin65
Area 1 =706.922
Area 2 = ½ ab SinC
Area 2 = ½ (38)(42) Sin123
Area 2 = ½ 1596 Sin123
Area 2 = 669.262
Total Area = Area 1 + Area 2
Total Area = 706.922 + 669.262
Total Area = 1376.22
Answers to the Quiz
1. a = b
SinA SinB
85 = 100
SinA Sin37
100SinA = 85Sin37
SinA = 85Sin37
100
A = Sin 0.5115
A = 31
C = 180 – 31 – 37 = 112
c2 = a2 + b2 – 2 ab CosC
c2 = 852 + 1002 – 2 (85)(100) Cos112
c2 = 7225 + 10000 – 2 (8500) Cos112
c2 = 17225 – 17000 Cos112
c2 = 17225 + 6368.3
c2 = 23593.3
c = 153.6 meters
2. x2 = b2 + c2 – 2 ab CosC
x2 = 4002 + 6002 – 2 (400)(600) Cos76
x2 = 160000 + 360000 – 2 (240000) Cos76
x2 = 520000 – 480000 Cos76
x2 = 520000 – 116122.51
x2 = 403877.5
x = 635.5 meters
c2 = a2 + b2 – 2 ab CosC
635.52 = 3002 + 5002 – 2 (300)(500) Cos
403860.25 = 90000 + 250000 – 2 (150000) CosC
403860.25 = 340000 - 300000 CosC
63860.25 = -300000CosC
-0.213 = CosC
C = Cos-1-0.213
C =102
Total Area = Area 1 + Area 2
Total Area = ½ ab SinC + ½ ab Sin C
Total Area = ½ (400)(600) Sin76 + ½ (300)(500) Sin102
Total Area = ½ 240000Sin76 + ½ 150000Sin102
Total Area = ½ 232871 + ½ 146722
Total Area = 116436 + 73361
Total Area = 1897972 meters
3. c2 = 1002 + 2002 – 2 (100)(200) Cos115
c2 = 10000 + 40000 – 2 (20000) Cos115
c2 = 50000 – 40000 Cos115
c2 = 17225 + 16905
c2 = 34130
c = 185 miles
Yes, there is a much shorter route home if they could drive straight back home from where they were it would only be 185 miles compared to 300 miles.
4.C = 180 – 61 – 43 = 76
a = c
SinA SinC
a = 80
Sin43 Sin76
aSin76 = 80Sin43
a = 80Sin43
Sin76
a = 56 miles
a = b
SinA SinB
56 = b
Sin43 Sin61
56Sin61 = bSin43
b = 56Sin61
Sin43
b = 72 miles
It would make more sense for Station A to send out a rescue vessel because they are only 56 miles from the shipwreck so it would be closer to save them.