Problem Set 58: Non-right Triangles: Law of Cosines

1. If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

2. If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?

3. Explain what [latex]s[/latex] represents in Heron’s formula.

4. Explain the relationship between the Pythagorean Theorem and the Law of Cosines.

5. When must you use the Law of Cosines instead of the Pythagorean Theorem?

For the following exercises, assume [latex]\alpha [/latex] is opposite side [latex]a,\beta [/latex] is opposite side [latex]b[/latex], and [latex]\gamma [/latex] is opposite side [latex]c[/latex]. If possible, solve each triangle for the unknown side. Round to the nearest tenth.

6. [latex]\gamma =41.2^\circ ,a=2.49,b=3.13[/latex]

7. [latex]\alpha =120^\circ ,b=6,c=7[/latex]

8. [latex]\beta =58.7^\circ ,a=10.6,c=15.7[/latex]

9. [latex]\gamma =115^\circ ,a=18,b=23[/latex]

10. [latex]\alpha =119^\circ ,a=26,b=14[/latex]

11. [latex]\gamma =113^\circ ,b=10,c=32[/latex]

12. [latex]\beta =67^\circ ,a=49,b=38[/latex]

13. [latex]\alpha =43.1^\circ ,a=184.2,b=242.8[/latex]

14. [latex]\alpha =36.6^\circ ,a=186.2,b=242.2[/latex]

15. [latex]\beta =50^\circ ,a=105,b=45{}_{}{}^{}[/latex]

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

16. [latex]a=42,b=19,c=30[/latex]; find angle [latex]A[/latex].

17. [latex]a=14,\text{ }b=13,\text{ }c=20[/latex]; find angle [latex]C[/latex].

18. [latex]a=16,b=31,c=20[/latex]; find angle [latex]B[/latex].

19. [latex]a=13,b=22,c=28[/latex]; find angle [latex]A[/latex].

20. [latex]a=108,b=132,c=160[/latex]; find angle [latex]C[/latex].

For the following exercises, solve the triangle. Round to the nearest tenth.

21. [latex]A=35^\circ ,b=8,c=11[/latex]

22. [latex]B=88^\circ ,a=4.4,c=5.2[/latex]

23. [latex]C=121^\circ ,a=21,b=37[/latex]

24. [latex]a=13,b=11,c=15[/latex]

25. [latex]a=3.1,b=3.5,c=5[/latex]

26. [latex]a=51,b=25,c=29[/latex]

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

27. Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

28. Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

29. [latex]a=\frac{1}{2}\text{m},b=\frac{1}{3}\text{m},c=\frac{1}{4}\text{m}[/latex]

30. [latex]a=12.4\text{ ft},\text{ }b=13.7\text{ ft},\text{ }c=20.2\text{ ft}[/latex]

31. [latex]a=1.6\text{ yd},\text{ }b=2.6\text{ yd},\text{ }c=4.1\text{ yd}[/latex]

For the following exercises, find the length of side [latex]x[/latex]. Round to the nearest tenth.

32.
A triangle. One angle is 72 degrees, with opposite side = x. The other two sides are 5 and 6.5.

33.
A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.

34.
A triangle. One angle is 40 degrees with opposite side = 15. The other two sides are 12 and x.

35.
A triangle. One angle is 65 degrees with opposite side = x. The other two sides are 30 and 23.

36.
A triangle. One angle is 50 degrees with opposite side = x. The other two sides are 225 and 305.

37.
A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

For the following exercises, find the measurement of angle [latex]A[/latex].

38.
A triangle. Angle A is opposite a side of length 2.3. The other two sides are 1.5 and 2.5.

39.
A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.

40.
A triangle. Angle A is opposite a side of length 6.8. The other two sides are 4.3 and 8.2.

41.
A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.

42. Find the measure of each angle in the triangle below. Round to the nearest tenth.
A triangle A B C. Angle A is opposite a side of length 10, angle B is opposite a side of length 12, and angle C is opposite a side of length 7.

For the following exercises, solve for the unknown side. Round to the nearest tenth.

43.
A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.

44.
A triangle. One angle is 30 degrees with opposite side unknown. The other two sides are 16 and 10.

45.
A triangle. One angle is 22 degrees with opposite side unknown. The other two sides are 20 and 13.

46.
A triangle. One angle is 88 degrees with opposite side = 9. Another side is 5.

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

47.
A triangle with sides 8, 12, and 17. Angles unknown.

48.
A triangle with sides 50, 22, and 36. Angles unknown.

49.
A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.

50.
A triangle with sides 8.9, 12.5, and 16.2. Angles unknown.

51.
A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

52. A parallelogram has sides of length 16 units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

53. The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

54. The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure of the larger angle is 100°. Find the length of the shorter diagonal.

55. A regular octagon is inscribed in a circle with a radius of 8 inches. Find the perimeter of the octagon.
An octagon inscribed in a circle.

56. A regular pentagon is inscribed in a circle of radius 12 cm. Find the perimeter of the pentagon. Round to the nearest tenth of a centimeter.
A pentagon inscribed in a circle.

For the following exercises, suppose that [latex]{x}^{2}=25+36 - 60\cos \left(52\right)[/latex] represents the relationship of three sides of a triangle and the cosine of an angle.

57. Draw the triangle.

58. Find the length of the third side.

For the following exercises, find the area of the triangle.

59.
A triangle. One angle is 22 degrees with opposite side = 3.4. Another side is 5.3.

60.
A triangle. One angle is 80 degrees with opposite side unknown. The other two sides are 8 and 6.

61.
A triangle. One angle is 18 degrees with opposite side = 12.8. Another side is 18.8.

62. A surveyor has taken the measurements shown in the triangle below. Find the distance across the lake. Round answers to the nearest tenth.
A triangle. One angle is 70 degrees with opposite side unknown, which is the length of the lake. The other two sides are 800 and 900 feet.

63. A satellite calculates the distances and angle shown in the diagram below (not to scale). Find the distance between the two cities. Round answers to the nearest tenth.
Insert figure(table) alt text: A triangle formed by two cities on the ground and a satellite above them. The angle by the satellite is 2.1 degrees with opposite side unknown, which is the distance between the two cities. The lengths of the other sides are 370 and 350 km.

64. An airplane flies 220 miles with a heading of 40°, and then flies 180 miles with a heading of 170°. How far is the plane from its starting point, and at what heading? Round answers to the nearest tenth.

65. A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in the image below. A guy-wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.
Two triangles, one on top of the other. The bottom triangle is the hill inclined 34 degrees to the horizontal. The second is formed by the base of the tower on the incline of the hill, the top of the tower, and the wire anchor point uphill from the tower on the incline. The sides are the tower, the incline of the hill, and the wire. The tower side is 113 feet and the incline side is 98 feet.

66. Two ships left a port at the same time. One ship traveled at a speed of 18 miles per hour at a heading of 320°. The other ship traveled at a speed of 22 miles per hour at a heading of 194°. Find the distance between the two ships after 10 hours of travel.

67. The graph in the figure below represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327° and the second boat is traveling at 4 miles per hour at a heading of 60°. Find the distance between the two boats after 2 hours.
Insert figure(table) alt text: A graph of two rays, which represent the paths of the two boats. Both rays start at the origin. The first goes into the first quadrant at a 60 degree angle at 4 mph. The second goes into the fourth quadrant at a 327 degree angle from the origin. The second travels at 18 mph.

68. A triangular swimming pool measures 40 feet on one side and 65 feet on another side. These sides form an angle that measures 50°. How long is the third side (to the nearest tenth)?

69. A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

70. Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.

71. Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

72. Two planes leave the same airport at the same time. One flies at 20° east of north at 500 miles per hour. The second flies at 30° east of south at 600 miles per hour. How far apart are the planes after 2 hours?

73. Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

74. A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 square units. Find the measure of the longer diagonal.

75. The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

76. The four sequential sides of a quadrilateral have lengths 5.7 cm, 7.2 cm, 9.4 cm, and 12.8 cm. The angle between the two smallest sides is 106°. What is the area of this quadrilateral?

77. Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.

78. Find the area of a triangular piece of land that measures 110 feet on one side and 250 feet on another; the included angle measures 85°. Round to the nearest whole square foot.