{"id":14094,"date":"2018-06-15T19:20:13","date_gmt":"2018-06-15T19:20:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/solution\/"},"modified":"2021-11-22T22:06:02","modified_gmt":"2021-11-22T22:06:02","slug":"solutions-law-of-sines","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solutions-law-of-sines\/","title":{"raw":"Solutions: Law of Sines","rendered":"Solutions: Law of Sines"},"content":{"raw":"

Solutions to Try Its<\/h2>\r\n1.\u00a0[latex]\\begin{array}{l}\\alpha ={98}^{\\circ }a=34.6\\\\ \\beta ={39}^{\\circ }b=22\\\\ \\gamma ={43}^{\\circ }c=23.8\\end{array}[\/latex]\r\n\r\n2.\r\nSolution 1<\/strong>\r\n[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]\r\nSolution 2<\/strong>\r\n[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]\r\n\r\n3. [latex]\\beta \\approx 5.7^\\circ ,\\gamma \\approx 94.3^\\circ ,c\\approx 101.3[\/latex]\r\n\r\n4. two\r\n\r\n5.\u00a0about\u00a08.2 square feet\r\n\r\n6.\u00a0161.9 yd\r\n

Solutions to Odd-Numbered Exercises<\/h2>\r\n1.\u00a0The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90\u00b0 angle.\r\n\r\n3.\u00a0When the known values are the side opposite the missing angle and another side and its opposite angle.\r\n\r\n5.\u00a0A triangle with two given sides and a non-included angle.\r\n\r\n7.\u00a0[latex] \\beta =72^\\circ ,a\\approx 12.0,b\\approx 19.9[\/latex]\r\n\r\n9.\u00a0[latex] \\gamma =20^\\circ ,b\\approx 4.5,c\\approx 1.6[\/latex]\r\n\r\n11.\u00a0[latex]b\\approx 3.78[\/latex]\r\n\r\n13.\u00a0[latex]c\\approx 13.70[\/latex]\r\n\r\n15.\u00a0one triangle, [latex]\\alpha \\approx 50.3^\\circ ,\\beta \\approx 16.7^\\circ ,a\\approx 26.7[\/latex]\r\n\r\n17.\u00a0two 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]\r\n\r\n19.\u00a0two 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]\r\n\r\n21.\u00a0two 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]\r\n\r\n23.\u00a0no triangle possible\r\n\r\n25.\u00a0[latex]A\\approx 47.8^\\circ [\/latex] or [latex]{A}^{\\prime }\\approx 132.2^\\circ [\/latex]\r\n\r\n27.\u00a0[latex]8.6[\/latex]\r\n\r\n29.\u00a0[latex]370.9[\/latex]\r\n\r\n31.\u00a0[latex]12.3[\/latex]\r\n\r\n33.\u00a0[latex]12.2 [\/latex]\r\n\r\n35.\u00a0[latex]16.0 [\/latex]\r\n\r\n37.\u00a0[latex]29.7^\\circ [\/latex]\r\n\r\n39.\u00a0[latex]x=76.9^\\circ \\text{or }x=103.1^\\circ [\/latex]\r\n\r\n41.\u00a0[latex]110.6^\\circ [\/latex]\r\n\r\n43.\u00a0[latex]A\\approx 39.4,\\text{ }C\\approx 47.6,\\text{ }BC\\approx 20.7 [\/latex]\r\n\r\n45.\u00a0[latex]57.1[\/latex]\r\n\r\n47.\u00a0[latex]42.0 [\/latex]\r\n\r\n49.\u00a0[latex]430.2 [\/latex]\r\n\r\n51.\u00a0[latex]10.1[\/latex]\r\n\r\n53.\u00a0[latex]AD\\approx \\text{ }13.8[\/latex]\r\n\r\n55.\u00a0[latex]AB\\approx 2.8 [\/latex]\r\n\r\n57.\u00a0[latex]L\\approx 49.7,\\text{ }N\\approx 56.3,\\text{ }LN\\approx 5.8[\/latex]\r\n\r\n59.\u00a051.4 feet\r\n\r\n61.\u00a0The distance from the satellite to station [latex]A[\/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.\r\n\r\n63.\u00a02.6\u00a0ft\r\n\r\n65.\u00a05.6\u00a0km\r\n\r\n67.\u00a0371\u00a0ft\r\n\r\n69.\u00a05936\u00a0ft\r\n\r\n71.\u00a024.1\u00a0ft\r\n\r\n73.\u00a019,056\u00a0ft2\r\n\r\n75.\u00a0445,624\u00a0square\u00a0miles\r\n\r\n77.\u00a08.65\u00a0ft2","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0[latex]\\begin{array}{l}\\alpha ={98}^{\\circ }a=34.6\\\\ \\beta ={39}^{\\circ }b=22\\\\ \\gamma ={43}^{\\circ }c=23.8\\end{array}[\/latex]<\/p>\n

2.
\nSolution 1<\/strong>
\n[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]
\nSolution 2<\/strong>
\n[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]<\/p>\n

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

4. two<\/p>\n

5.\u00a0about\u00a08.2 square feet<\/p>\n

6.\u00a0161.9 yd<\/p>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90\u00b0 angle.<\/p>\n

3.\u00a0When the known values are the side opposite the missing angle and another side and its opposite angle.<\/p>\n

5.\u00a0A triangle with two given sides and a non-included angle.<\/p>\n

7.\u00a0[latex]\\beta =72^\\circ ,a\\approx 12.0,b\\approx 19.9[\/latex]<\/p>\n

9.\u00a0[latex]\\gamma =20^\\circ ,b\\approx 4.5,c\\approx 1.6[\/latex]<\/p>\n

11.\u00a0[latex]b\\approx 3.78[\/latex]<\/p>\n

13.\u00a0[latex]c\\approx 13.70[\/latex]<\/p>\n

15.\u00a0one triangle, [latex]\\alpha \\approx 50.3^\\circ ,\\beta \\approx 16.7^\\circ ,a\\approx 26.7[\/latex]<\/p>\n

17.\u00a0two 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]<\/p>\n

19.\u00a0two 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]<\/p>\n

21.\u00a0two 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]<\/p>\n

23.\u00a0no triangle possible<\/p>\n

25.\u00a0[latex]A\\approx 47.8^\\circ[\/latex] or [latex]{A}^{\\prime }\\approx 132.2^\\circ[\/latex]<\/p>\n

27.\u00a0[latex]8.6[\/latex]<\/p>\n

29.\u00a0[latex]370.9[\/latex]<\/p>\n

31.\u00a0[latex]12.3[\/latex]<\/p>\n

33.\u00a0[latex]12.2[\/latex]<\/p>\n

35.\u00a0[latex]16.0[\/latex]<\/p>\n

37.\u00a0[latex]29.7^\\circ[\/latex]<\/p>\n

39.\u00a0[latex]x=76.9^\\circ \\text{or }x=103.1^\\circ[\/latex]<\/p>\n

41.\u00a0[latex]110.6^\\circ[\/latex]<\/p>\n

43.\u00a0[latex]A\\approx 39.4,\\text{ }C\\approx 47.6,\\text{ }BC\\approx 20.7[\/latex]<\/p>\n

45.\u00a0[latex]57.1[\/latex]<\/p>\n

47.\u00a0[latex]42.0[\/latex]<\/p>\n

49.\u00a0[latex]430.2[\/latex]<\/p>\n

51.\u00a0[latex]10.1[\/latex]<\/p>\n

53.\u00a0[latex]AD\\approx \\text{ }13.8[\/latex]<\/p>\n

55.\u00a0[latex]AB\\approx 2.8[\/latex]<\/p>\n

57.\u00a0[latex]L\\approx 49.7,\\text{ }N\\approx 56.3,\\text{ }LN\\approx 5.8[\/latex]<\/p>\n

59.\u00a051.4 feet<\/p>\n

61.\u00a0The distance from the satellite to station [latex]A[\/latex] is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.<\/p>\n

63.\u00a02.6\u00a0ft<\/p>\n

65.\u00a05.6\u00a0km<\/p>\n

67.\u00a0371\u00a0ft<\/p>\n

69.\u00a05936\u00a0ft<\/p>\n

71.\u00a024.1\u00a0ft<\/p>\n

73.\u00a019,056\u00a0ft2<\/p>\n

75.\u00a0445,624\u00a0square\u00a0miles<\/p>\n

77.\u00a08.65\u00a0ft2<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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