Section Exercises 4.5: Non-Right Triangles: Law of Sines

1. Describe the altitude of a triangle.

2. Compare right triangles and oblique triangles.

3. When can you use the Law of Sines to find a missing angle?

4. In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

5. What type of triangle results in an ambiguous case?

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]. Solve each triangle, if possible. Round each answer to the nearest tenth.

6. [latex]\alpha =43^\circ ,\gamma =69^\circ ,a=20[/latex]

7. [latex]\alpha =35^\circ ,\gamma =73^\circ ,c=20[/latex]

8. [latex]\alpha =60^\circ ,\beta =60^\circ ,\gamma =60^\circ[/latex]

9. [latex]a=4,\alpha =60^\circ ,\beta =100^\circ[/latex]

10. [latex]b=10,\beta =95^\circ ,\gamma =30^\circ[/latex]

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle [latex]A[/latex] is opposite side [latex]a[/latex], angle [latex]B[/latex] is opposite side [latex]b[/latex], and angle [latex]C[/latex] is opposite side [latex]c[/latex].

11. Find side [latex]b[/latex] when [latex]A=37^\circ ,B=49^\circ ,c=5[/latex].

12. Find side [latex]a[/latex] when [latex]A=132^\circ ,C=23^\circ ,b=10[/latex].

13. Find side [latex]c[/latex] when [latex]B=37^\circ ,C=21,b=23[/latex].

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]. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

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

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

16. [latex]b=3.5,c=5.3,\gamma =80^\circ[/latex]

17. [latex]a=12,c=17,\alpha =35^\circ[/latex]

18. [latex]a=20.5,b=35.0,\beta =25^\circ[/latex]

19. [latex]a=7,c=9,\alpha =43^\circ[/latex]

20. [latex]a=7,b=3,\beta =24^\circ[/latex]

21. [latex]b=13,c=5,\gamma =10^\circ[/latex]

22. [latex]a=2.3,c=1.8,\gamma =28^\circ[/latex]

23. [latex]\beta =119^\circ ,b=8.2,a=11.3[/latex]

For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

24. Find angle [latex]A[/latex] when [latex]a=24,b=5,B=22^\circ[/latex].

25. Find angle [latex]A[/latex] when [latex]a=13,b=6,B=20^\circ[/latex].

26. Find angle [latex]B[/latex] when [latex]A=12^\circ ,a=2,b=9[/latex].

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

27. [latex]a=5,c=6,\beta =35^\circ[/latex]

28. [latex]b=11,c=8,\alpha =28^\circ[/latex]

29. [latex]a=32,b=24,\gamma =75^\circ[/latex]

30. [latex]a=7.2,b=4.5,\gamma =43^\circ[/latex]

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

31.
A triangle with an angle of 50 degrees and opposite side of length 10. Another angle is 70 degrees with side opposite of length x.

32.
A triangle with one angle = 120 degrees. Another angle is 25 degrees with side opposite = x. The side adjacent to the 25 and 120 degree angles is of length 6.

33.
A triangle. One angle is 45 degrees with side opposite = x. Another angle is 75 degrees. The side adjacent to the 45 and 75 degree angles = 15.

34.
A triangle. One angle is 40 degrees with opposite side = x. Another angle is 110 degrees with side opposite = 18.

35.
A triangle. One angle is 50 degrees with opposite = x. Another angle is 42 degrees with opposite side = 14.

36.
A triangle. One angle is 111 degrees with opposite side = x. Another angle is 22 degrees. The side adjacent to the 111 and 22 degree angles = 8.6.

For the following exercises, find the measure of angle [latex]x[/latex], if possible. Round to the nearest tenth.

37.
A triangle. One angles is 98 degrees with opposite side = 10. Another angle is x degrees with opposite side = 5.

38.
A triangle. One angle is 37 degrees with opposite side = 11. Another angle is x degrees with opposite side = 8.

39.
A triangle. One angle is 22 degrees with side opposite = 5. Another angle is x degrees with opposite side = 13.

40.
A triangle. One angle is 59 degrees with opposite side = 5.7. Another angle is x degrees with opposite side = 5.3.

41. Notice that [latex]x[/latex] is an obtuse angle.
A triangle. One angle is 55 degrees with side opposite = 21. Another angle is x degrees with opposite side = 24.

42.
A triangle. One angle is 65 degrees with opposite side = 10. Another angle is x degrees with opposite side = 12.

For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.

43.
A triangle. One angle is 93 degrees with opposite side = 32.6. Another side is 24.1.

44.
A triangle. One angle is 30 degrees. The two sides adjacent to that angle are 10 and 16.

45.
A triangle. One angle is 25 degrees. The two sides adjacent to that angle are 18 and 15

46.
A triangle. One angle is 51 degrees with opposite side = 3.5. The other two sides are 4.5 and 2.9.

47.
A triangle. One angle is 58 degrees with opposite side unknown. Another angle is 51 degrees with opposite side = 9. The side adjacent to the two given angles is 11.

48.
A triangle. One angle is 40 degrees with opposite side = 18. One of the other sides is 25.

49.
A triangle. One angle is 115 degrees with opposite side = 50. Another angle is 30 degrees with opposite side = 30.

50. Find the radius of the circle below. Round to the nearest tenth.
A triangle inscribed in a circle. Two of the legs are radii. The central angle formed by the radii is 145 degrees, and the opposite side is 3.

51. Find the diameter of the circle below. Round to the nearest tenth.
A triangle inscribed in a circle. Two of the legs are radii. The central angle formed by the radii is 110 degrees, and the opposite side is 8.3.

52. Find [latex]m\angle ADC[/latex] in the figure below. Round to the nearest tenth.
A triangle inside a triangle. The outer triangle is formed by vertices A, B, and D. Side B D is the base. The inner triangle shares vertices A and B. The last vertex C is located on the base side of the outer triangle between vertices B and D. Angle B is 60 degrees, side A D is 10, and side A C is 9.

53. Find [latex]AD[/latex] in the triangle below. Round to the nearest tenth.
A triangle inside a triangle. The outer triangle is formed by vertices A, B, and D. Side B D is the base. The inner triangle shares vertices A and B. The last vertex C is located on the base side of the outer triangle between vertices B and D. Angle B is 53 degrees, angle D is 44 degrees, side A B is 12, and side A C is 13.

54. Solve both triangles. Round each answer to the nearest tenth.
Two triangles formed by intersecting lines A D and B C. They intersect at point E. The first triangle is formed from vertices A, B, and E while the second triangle is formed from vertices C, E, and D. Angle A is 48 degrees, side A B is 4.2, angle D is 48 degrees, and side C D is 2. Angle A E B is 46 degrees.

55. Find [latex]AB[/latex] in the parallelogram below.
A parallelogram with vertices A, B, C, and D. There is a diagonal from vertex B to vertex C. Angle A is 130 degrees, angle D is 130 degrees, side B D is 10, and the diagonal B C is 12.

56.  Solve the triangle below. (Hint: Draw a perpendicular from [latex]H[/latex] to [latex]JK[/latex]). Round each answer to the nearest tenth.
A triangle with vertices J, K, and H. Side J K is the horizontal base and is 10. Side JH is 7. Angle J is 20 degrees.

57. Solve the triangle below. (Hint: Draw a perpendicular from [latex]N[/latex] to [latex]LM[/latex]). Round each answer to the nearest tenth.
A triangle with vertices M, N, and L. Side M N is the horizontal base and is 4.6. Angle M is 74 degrees, and side M L is 5.

58. In the figure below, [latex]ABCD[/latex] is not a parallelogram. [latex]\angle m[/latex] is obtuse. Solve both triangles. Round each answer to the nearest tenth.
A quadrilateral with vertices A, B, C, and D. There is a diagonal from vertex B to vertex D of length 45. Side A B is x, side B C is y, side C D is 40, and side D A is 29. Angle A is m degrees, angle C is 65 degrees, angle A B D is 35 degrees, angle D B C is n degrees, angle B D C is k degrees, and angle A D B is h degrees.

59. A pole leans away from the sun at an angle of [latex]7^\circ[/latex] to the vertical, as shown in below. When the elevation of the sun is [latex]55^\circ[/latex], the pole casts a shadow 42 feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.
A triangle within a triangle. The outer triangle is formed by vertices A, B, and S (the sun). Side A B is the horizontal base, the ground, and is 42 feet. Angle A is 55 degrees. The inner triangle is formed by vertices A, B, and C. Side B C is the pole. Vertex C is located on side A S of the outer triangle between vertices A and S. Angle C B S is 7 degrees.

60. To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown below. Determine the distance of the boat from station [latex]A[/latex] and the distance of the boat from shore. Round your answers to the nearest whole foot.
A triangle formed by the two radar stations A and B and the boat. Side A B is the horizontal base. Angle A is 70 degrees and angle B is 60 degrees.

61. The diagram below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations [latex]A[/latex] and [latex]B[/latex], which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at [latex]A[/latex] and [latex]B[/latex] are measured to be [latex]86.2^\circ[/latex] and [latex]83.9^\circ[/latex], respectively. How far is the satellite from station [latex]A[/latex] and how high is the satellite above the ground? Round answers to the nearest whole mile.
A triangle formed by two ground tracking stations A and B and the satellite. Side A B is the horizontal base of the triangle. Angle A is 83.9 degrees, and the supplementary angle to angle B is 86.2 degrees.

62. A communications tower is located at the top of a steep hill, as shown below. The angle of inclination of the hill is [latex]67^\circ[/latex]. A guy wire is to be attached to the top of the tower and to the ground, 165 meters downhill from the base of the tower. The angle formed by the guy wire and the hill is [latex]16^\circ[/latex]. Find the length of the cable required for the guy wire to the nearest whole meter.
A triangle formed by the bottom of the hill, the base of the tower at the top of the hill, and the top of the tower. The side between the bottom of the hill and the top of the tower is wire. The length of the side bertween the bottom of the hill and the bottom of the tower is 165 meters. The angle formed by the wire side and the bottom of the hill is 16 degrees. The angle between the hill and the horizontal ground is 67 degrees.

63. The roof of a house is at a [latex]20^\circ[/latex] angle. An 8-foot solar panel is to be mounted on the roof and should be angled [latex]38^\circ[/latex] relative to the horizontal for optimal results. How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.
A triangle whose sides are the solar panel, the roof which goes past the solar panel, and the vertical support for the panel. The solar panel side is 8 feet long. There are horizontal dotted lines at the bottom of the solar panel and the bottom of the roof. The angle between the solar panel and the horizontal is 38 degrees. The angle between the roof and the horizontal is 20 degrees.

64. Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.6 km apart, to be [latex]37^\circ[/latex] and [latex]44^\circ[/latex], as shown below. Find the distance of the plane from point [latex]A[/latex] to the nearest tenth of a kilometer.
A triangle formed by points A and B on the ground and a plane in the air between them. Side A B is the horizontal ground. There is a horizontal dotted line parallel to the ground going through the plane. The angle formed by the dotted horizontal, the plane, and point A is 37 degrees. The angle between the dotted horizontal, the plane, and point B is 44 degrees.

65. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32° and 56°, as shown below. Find the distance of the plane from point [latex]A[/latex] to the nearest tenth of a kilometer.
A triangle formed between the plane and two points on the ground, A and B. Side A B is the horizontal base. The plane is above and to the left of both A and B. Point B is to the right of point A. There is a dotted horizontal line going through the plane parallel to the ground. The angle formed between point B, the plane, and the dotted horizontal line is 32 degrees. The angle formed between point A, the plane, and the dotted horizontal line is 56 degrees.

66. In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 39°. They then move 300 feet closer to the building and find the angle of elevation to be 50°. Assuming that the street is level, estimate the height of the building to the nearest foot.

67. In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the nearest foot.

68. Points [latex]A[/latex] and [latex]B[/latex] are on opposite sides of a lake. Point [latex]C[/latex] is 97 meters from [latex]A[/latex]. The measure of angle [latex]BAC[/latex] is determined to be 101°, and the measure of angle [latex]ACB[/latex] is determined to be 53°. What is the distance from [latex]A[/latex] to [latex]B[/latex], rounded to the nearest whole meter?

69. A man and a woman standing [latex]3\frac{1}{2}[/latex] miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27°, and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon to the nearest foot.

70. Two search teams spot a stranded climber on a mountain. The first search team is 0.5 miles from the second search team, and both teams are at an altitude of 1 mile. The angle of elevation from the first search team to the stranded climber is 15°. The angle of elevation from the second search team to the climber is 22°. What is the altitude of the climber? Round to the nearest tenth of a mile.

71. A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28°. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28°. If the man and woman are 20 feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

72. Three cities, [latex]A,B[/latex], and [latex]C[/latex], are located so that city [latex]A[/latex] is due east of city [latex]B[/latex]. If city [latex]C[/latex] is located 35° west of north from city [latex]B[/latex] and is 100 miles from city [latex]A[/latex] and 70 miles from city [latex]B[/latex], how far is city [latex]A[/latex] from city [latex]B?[/latex] Round the distance to the nearest tenth of a mile.

73. Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

74. Brian’s house is on a corner lot. Find the area of the front yard if the edges measure 40 and 56 feet.
A triangle with angle 135 degrees. The sides adjacent to that angle are 56 feet and 40 feet. The other side is the house, length unknown.

75. The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62°.

76. A yield sign measures 30 inches on all three sides. What is the area of the sign?

77. Naomi bought a modern dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32° and 42°.
A triangle. One angle is 32 degrees with opposite side = 4. Another angle is 42 degrees with opposite side = 4.5.