The Law of Sines and Cosines.

 

 

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.

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)

ÐA = 35°

 

ÐC = 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

    ÐA = 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

 

ÐC = 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)

ÐC = 59°

 

ÐA = 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. ÐA = 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

ÐB = 21°

 

ÐC = 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

ÐA = 40°

 

ÐC = 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

ÐC = 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

ÐA = 31°

 

ÐC = 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. ÐC = 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.

 

 

 

experimental group