Solutions for Non-Right Triangles: Law of Sines

Solutions to Try Its

1. [latex]\begin{array}{l}\alpha ={98}^{\circ }a=34.6\\ \beta ={39}^{\circ }b=22\\ \gamma ={43}^{\circ }c=23.8\end{array}[/latex]

Solution 1
[latex]\begin{array}{ll}\alpha =80^\circ \hfill & a=120\hfill \\ \beta \approx 83.2^\circ \hfill & b=121\hfill \\ \gamma \approx 16.8^\circ \hfill & c\approx 35.2\hfill \end{array}[/latex]
Solution 2
[latex]\begin{array}{l}{\alpha }^{\prime }=80^\circ {a}^{\prime }=120\hfill \\ {\beta }^{\prime }\approx 96.8^\circ {b}^{\prime }=121\hfill \\ {\gamma }^{\prime }\approx 3.2^\circ {c}^{\prime }\approx 6.8\hfill \end{array}[/latex]

3. [latex]\beta \approx 5.7^\circ ,\gamma \approx 94.3^\circ ,c\approx 101.3[/latex]

4. two

5. about 8.2 square feet

6. 161.9 yd

Solutions to Odd-Numbered Exercises

1. The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

3. When the known values are the side opposite the missing angle and another side and its opposite angle.

5. A triangle with two given sides and a non-included angle.

7. [latex] \beta =72^\circ ,a\approx 12.0,b\approx 19.9[/latex]

9. [latex] \gamma =20^\circ ,b\approx 4.5,c\approx 1.6[/latex]

11. [latex]b\approx 3.78[/latex]

13. [latex]c\approx 13.70[/latex]

15. one triangle, [latex]\alpha \approx 50.3^\circ ,\beta \approx 16.7^\circ ,a\approx 26.7[/latex]

17. two triangles, [latex] \gamma \approx 54.3^\circ ,\beta \approx 90.7^\circ ,b\approx 20.9[/latex] or [latex] {\gamma }^{\prime }\approx 125.7^\circ ,{\beta }^{\prime }\approx 19.3^\circ ,{b}^{\prime }\approx 6.9[/latex]

19. two triangles, [latex] \beta \approx 75.7^\circ , \gamma \approx 61.3^\circ ,b\approx 9.9[/latex] or [latex] {\beta }^{\prime }\approx 18.3^\circ ,{\gamma }^{\prime }\approx 118.7^\circ ,{b}^{\prime }\approx 3.2[/latex]

21. two triangles, [latex]\alpha \approx 143.2^\circ ,\beta \approx 26.8^\circ ,a\approx 17.3[/latex] or [latex]{\alpha }^{\prime }\approx 16.8^\circ ,{\beta }^{\prime }\approx 153.2^\circ ,{a}^{\prime }\approx 8.3[/latex]

23. no triangle possible

25. [latex]A\approx 47.8^\circ [/latex] or [latex]{A}^{\prime }\approx 132.2^\circ [/latex]

27. [latex]8.6[/latex]

29. [latex]370.9[/latex]

31. [latex]12.3[/latex]

33. [latex]12.2 [/latex]

35. [latex]16.0 [/latex]

37. [latex]29.7^\circ [/latex]

39. [latex]x=76.9^\circ \text{or }x=103.1^\circ [/latex]

41. [latex]110.6^\circ [/latex]

43. [latex]A\approx 39.4,\text{ }C\approx 47.6,\text{ }BC\approx 20.7 [/latex]

45. [latex]57.1[/latex]

47. [latex]42.0 [/latex]

49. [latex]430.2 [/latex]

51. [latex]10.1[/latex]

53. [latex]AD\approx \text{ }13.8[/latex]

55. [latex]AB\approx 2.8 [/latex]

57. [latex]L\approx 49.7,\text{ }N\approx 56.3,\text{ }LN\approx 5.8[/latex]

59. 51.4 feet

61. The distance from the satellite to station [latex]A[/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

63. 2.6 ft

65. 5.6 km

67. 371 ft

69. 5936 ft

71. 24.1 ft

73. 19,056 ft2

75. 445,624 square miles

77. 8.65 ft2