{"id":344,"date":"2026-02-16T20:55:08","date_gmt":"2026-02-16T20:55:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=344"},"modified":"2026-08-07T20:03:51","modified_gmt":"2026-08-07T20:03:51","slug":"triangle-trigonometry-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/triangle-trigonometry-get-stronger-answer-key\/","title":{"raw":"Triangle Trigonometry: Get Stronger Answer Key","rendered":"Triangle Trigonometry: Get Stronger Answer Key"},"content":{"raw":"

Right Triangle Trigonometry<\/h1>\r\n1.\r\n\"A\r\n\r\n2. The two acute angles are complementary.\r\n\r\n3. [latex]\\frac{\\pi }{6}[\/latex]\r\n\r\n4. [latex]\\frac{\\pi }{4}[\/latex]\r\n\r\n5. [latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]\r\n\r\n6. [latex]a=10,000,c=10,000.5[\/latex]\r\n\r\n7. [latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]\r\n\r\n8. [latex]\\frac{5\\sqrt{29}}{29}[\/latex]\r\n\r\n9. [latex]\\frac{5}{2}[\/latex]\r\n\r\n10. [latex]\\frac{\\sqrt{29}}{2}[\/latex]\r\n\r\n11. [latex]\\frac{5\\sqrt{41}}{41}[\/latex]\r\n\r\n12. [latex]\\frac{5}{4}[\/latex]\r\n\r\n13. [latex]\\frac{\\sqrt{41}}{4}[\/latex]\r\n\r\n14. [latex]c=14, b=7\\sqrt{3}[\/latex]\r\n\r\n15. [latex]a=15, b=15[\/latex]\r\n\r\n16. [latex]b=9.9970, c=12.2041[\/latex]\r\n\r\n17. [latex]a=2.0838, b=11.8177[\/latex]\r\n\r\n18. [latex]a=55.9808,c=57.9555[\/latex]\r\n\r\n19. [latex]a=46.6790,b=17.9184[\/latex]\r\n\r\n20. [latex]a=16.4662,c=16.8341[\/latex]\r\n\r\n21. 188.3159\r\n\r\n22. 200.6737\r\n\r\n23. 498.3471 ft\r\n\r\n24. 1060.09 ft\r\n\r\n25. 27.372 ft\r\n\r\n26. 22.6506 ft\r\n\r\n27. 368.7633 ft\r\n

Inverse Trigonometric Functions<\/h1>\r\n1. The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].\r\n\r\n2. Rewrite the expression as: [latex]\\sec^{-1}(2)=\\cos^{-1}!\\left(\\frac{1}{2}\\right)[\/latex]. Then use the inverse cosine (arccos) function on the calculator: [latex]\\theta=\\cos^{-1}\\left(\\frac{1}{2}\\right)[\/latex]\r\n\r\n3. [latex]\u2212\\frac{\\pi}{6}[\/latex]\r\n\r\n4. [latex]\\frac{3\\pi}{4}[\/latex]\r\n\r\n5. [latex]\\frac{\\pi}{3}[\/latex]\r\n\r\n6. 1.98\r\n\r\n7. 0.93\r\n\r\n8. 1.41\r\n\r\n9. 0.56 radians\r\n\r\n10. 0\r\n\r\n11. \u22120.71\r\n\r\n12. [latex]\u2212\\frac{\\pi}{4}[\/latex]\r\n\r\n13. 0.8\r\n\r\n14. [latex]\\frac{5}{13}[\/latex]\r\n\r\n15. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]\r\n\r\n16. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]\r\n\r\n17. t\r\n\r\n18. domain [\u22121,1]; range [0,\u03c0]\r\n\"A\r\n\r\n19. 0.395 radians\r\n\r\n20. 1.11 radians\r\n\r\n21. 1.25 radians\r\n\r\n22. 0.405 radians\r\n

Non-Right Triangles: Law of Sines<\/h1>\r\n1. When the known values are the side opposite the missing angle and another side and its opposite angle.\r\n\r\n2. A triangle with two given sides and a non-included angle.\r\n\r\n3. [latex] \\beta =72^\\circ ,a\\approx 12.0,b\\approx 19.9[\/latex]\r\n\r\n4. [latex] \\gamma =20^\\circ ,b\\approx 4.5,c\\approx 1.6[\/latex]\r\n\r\n5. [latex]b\\approx 3.78[\/latex]\r\n\r\n6. [latex]c\\approx 13.70[\/latex]\r\n\r\n7. one triangle, [latex]\\alpha \\approx 50.3^\\circ ,\\beta \\approx 16.7^\\circ ,a\\approx 26.7[\/latex]\r\n\r\n8. 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]\r\n\r\n9. 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]\r\n\r\n10. 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]\r\n\r\n11. no triangle possible\r\n\r\n12. [latex]A\\approx 47.8^\\circ [\/latex] or [latex]{A}^{\\prime }\\approx 132.2^\\circ [\/latex]\r\n\r\n13. [latex]8.6[\/latex]\r\n\r\n14. [latex]370.9[\/latex]\r\n\r\n15. [latex]12.3[\/latex]\r\n\r\n16. [latex]12.2 [\/latex]\r\n\r\n17. [latex]16.0 [\/latex]\r\n\r\n18. [latex]29.7^\\circ [\/latex]\r\n\r\n19. [latex]x=76.9^\\circ \\text{or }x=103.1^\\circ [\/latex]\r\n\r\n20. [latex]110.6^\\circ [\/latex]\r\n\r\n21. [latex]A\\approx 39.4,\\text{ }C\\approx 47.6,\\text{ }BC\\approx 20.7 [\/latex]\r\n\r\n22. [latex]57.1[\/latex]\r\n\r\n23. [latex]42.0 [\/latex]\r\n\r\n24. [latex]430.2 [\/latex]\r\n\r\n25. 51.4 feet\r\n\r\n26. The 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\n27. 2.6 ft\r\n\r\n28. 5.6 km\r\n\r\n29. 371 ft\r\n\r\n30. 24.1 ft\r\n\r\n31. 19,056 ft2\r\n\r\n32. 445,624 square miles\r\n

Non-Right Triangles: Law of Cosines<\/h1>\r\n1. two sides and the angle opposite the missing side\r\n\r\n[new item 2, \"what do you need to know for a missing angle,\" has no matching answer in this key]\r\n\r\n2. [latex]s[\/latex] is the semi-perimeter, which is half the perimeter of the triangle.\r\n\r\n[unmatched: key answer \"The Law of Cosines must be used for any oblique (non-right) triangle\" doesn't correspond to any problem in the exercise list]\r\n\r\n3. 11.3\r\n\r\n4. 257.4\r\n\r\n5. not possible\r\n\r\n6. 95.5\u00b0\r\n\r\n7. 26.9\u00b0\r\n\r\n8. [latex]B\\approx 45.9^\\circ ,C\\approx 99.1^\\circ ,a\\approx 6.4[\/latex]\r\n\r\n9. [latex]A\\approx 20.6^\\circ ,B\\approx 38.4^\\circ ,c\\approx 51.1[\/latex]\r\n\r\n10. [latex]A\\approx 37.8^\\circ ,B\\approx 43.8,C\\approx 98.4^\\circ [\/latex]\r\n\r\n11. 177.56 in2\r\n\r\n12. 0.04 m2\r\n\r\n13. 0.91 yd2\r\n\r\n14. 3.0\r\n\r\n15. 0.5\r\n\r\n16. 70.7\u00b0\r\n\r\n17. 77.4\u00b0\r\n\r\n18. 25.0\r\n\r\n19. 43.52\r\n\r\n20. 1.41\r\n\r\n21. 0.14\r\n\r\n22. 7.62\r\n\r\n23. 85.1\r\n\r\n24. 24.0 km\r\n\r\n25. 99.9 ft\r\n\r\n26. 37.3 miles\r\n\r\n27. 2371 miles\r\n\r\n28.\r\n\"Angle\r\n\r\n29. 599.8 miles\r\n\r\n30. 65.4 cm2\r\n\r\n31. 468 ft2","rendered":"

Right Triangle Trigonometry<\/h1>\n

1.
\n\"A<\/p>\n

2. The two acute angles are complementary.<\/p>\n

3. [latex]\\frac{\\pi }{6}[\/latex]<\/p>\n

4. [latex]\\frac{\\pi }{4}[\/latex]<\/p>\n

5. [latex]b=\\frac{20\\sqrt{3}}{3},c=\\frac{40\\sqrt{3}}{3}[\/latex]<\/p>\n

6. [latex]a=10,000,c=10,000.5[\/latex]<\/p>\n

7. [latex]b=\\frac{5\\sqrt{3}}{3},c=\\frac{10\\sqrt{3}}{3}[\/latex]<\/p>\n

8. [latex]\\frac{5\\sqrt{29}}{29}[\/latex]<\/p>\n

9. [latex]\\frac{5}{2}[\/latex]<\/p>\n

10. [latex]\\frac{\\sqrt{29}}{2}[\/latex]<\/p>\n

11. [latex]\\frac{5\\sqrt{41}}{41}[\/latex]<\/p>\n

12. [latex]\\frac{5}{4}[\/latex]<\/p>\n

13. [latex]\\frac{\\sqrt{41}}{4}[\/latex]<\/p>\n

14. [latex]c=14, b=7\\sqrt{3}[\/latex]<\/p>\n

15. [latex]a=15, b=15[\/latex]<\/p>\n

16. [latex]b=9.9970, c=12.2041[\/latex]<\/p>\n

17. [latex]a=2.0838, b=11.8177[\/latex]<\/p>\n

18. [latex]a=55.9808,c=57.9555[\/latex]<\/p>\n

19. [latex]a=46.6790,b=17.9184[\/latex]<\/p>\n

20. [latex]a=16.4662,c=16.8341[\/latex]<\/p>\n

21. 188.3159<\/p>\n

22. 200.6737<\/p>\n

23. 498.3471 ft<\/p>\n

24. 1060.09 ft<\/p>\n

25. 27.372 ft<\/p>\n

26. 22.6506 ft<\/p>\n

27. 368.7633 ft<\/p>\n

Inverse Trigonometric Functions<\/h1>\n

1. The function [latex]y=\\sin x[\/latex] is one-to-one on [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]; thus, this interval is the range of the inverse function of [latex]y=\\sin x\\text{, }f\\left(x\\right)=\\sin^{\u22121}x[\/latex]. The function [latex]y=\\cos x[\/latex] is one-to-one on [0,\u03c0]; thus, this interval is the range of the inverse function of [latex]y=\\cos x\\text{, }f(x)=\\cos^{\u22121}x[\/latex].<\/p>\n

2. Rewrite the expression as: [latex]\\sec^{-1}(2)=\\cos^{-1}!\\left(\\frac{1}{2}\\right)[\/latex]. Then use the inverse cosine (arccos) function on the calculator: [latex]\\theta=\\cos^{-1}\\left(\\frac{1}{2}\\right)[\/latex]<\/p>\n

3. [latex]\u2212\\frac{\\pi}{6}[\/latex]<\/p>\n

4. [latex]\\frac{3\\pi}{4}[\/latex]<\/p>\n

5. [latex]\\frac{\\pi}{3}[\/latex]<\/p>\n

6. 1.98<\/p>\n

7. 0.93<\/p>\n

8. 1.41<\/p>\n

9. 0.56 radians<\/p>\n

10. 0<\/p>\n

11. \u22120.71<\/p>\n

12. [latex]\u2212\\frac{\\pi}{4}[\/latex]<\/p>\n

13. 0.8<\/p>\n

14. [latex]\\frac{5}{13}[\/latex]<\/p>\n

15. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]<\/p>\n

16. [latex]\\frac{\\sqrt{2x+1}}{x+1}[\/latex]<\/p>\n

17. t<\/p>\n

18. domain [\u22121,1]; range [0,\u03c0]
\n\"A<\/p>\n

19. 0.395 radians<\/p>\n

20. 1.11 radians<\/p>\n

21. 1.25 radians<\/p>\n

22. 0.405 radians<\/p>\n

Non-Right Triangles: Law of Sines<\/h1>\n

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

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

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

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

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

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

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

8. 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]<\/p>\n

9. 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]<\/p>\n

10. 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]<\/p>\n

11. no triangle possible<\/p>\n

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

13. [latex]8.6[\/latex]<\/p>\n

14. [latex]370.9[\/latex]<\/p>\n

15. [latex]12.3[\/latex]<\/p>\n

16. [latex]12.2[\/latex]<\/p>\n

17. [latex]16.0[\/latex]<\/p>\n

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

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

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

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

22. [latex]57.1[\/latex]<\/p>\n

23. [latex]42.0[\/latex]<\/p>\n

24. [latex]430.2[\/latex]<\/p>\n

25. 51.4 feet<\/p>\n

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

27. 2.6 ft<\/p>\n

28. 5.6 km<\/p>\n

29. 371 ft<\/p>\n

30. 24.1 ft<\/p>\n

31. 19,056 ft2<\/p>\n

32. 445,624 square miles<\/p>\n

Non-Right Triangles: Law of Cosines<\/h1>\n

1. two sides and the angle opposite the missing side<\/p>\n

[new item 2, “what do you need to know for a missing angle,” has no matching answer in this key]<\/p>\n

2. [latex]s[\/latex] is the semi-perimeter, which is half the perimeter of the triangle.<\/p>\n

[unmatched: key answer “The Law of Cosines must be used for any oblique (non-right) triangle” doesn’t correspond to any problem in the exercise list]<\/p>\n

3. 11.3<\/p>\n

4. 257.4<\/p>\n

5. not possible<\/p>\n

6. 95.5\u00b0<\/p>\n

7. 26.9\u00b0<\/p>\n

8. [latex]B\\approx 45.9^\\circ ,C\\approx 99.1^\\circ ,a\\approx 6.4[\/latex]<\/p>\n

9. [latex]A\\approx 20.6^\\circ ,B\\approx 38.4^\\circ ,c\\approx 51.1[\/latex]<\/p>\n

10. [latex]A\\approx 37.8^\\circ ,B\\approx 43.8,C\\approx 98.4^\\circ[\/latex]<\/p>\n

11. 177.56 in2<\/p>\n

12. 0.04 m2<\/p>\n

13. 0.91 yd2<\/p>\n

14. 3.0<\/p>\n

15. 0.5<\/p>\n

16. 70.7\u00b0<\/p>\n

17. 77.4\u00b0<\/p>\n

18. 25.0<\/p>\n

19. 43.52<\/p>\n

20. 1.41<\/p>\n

21. 0.14<\/p>\n

22. 7.62<\/p>\n

23. 85.1<\/p>\n

24. 24.0 km<\/p>\n

25. 99.9 ft<\/p>\n

26. 37.3 miles<\/p>\n

27. 2371 miles<\/p>\n

28.
\n\"Angle<\/p>\n

29. 599.8 miles<\/p>\n

30. 65.4 cm2<\/p>\n

31. 468 ft2<\/p>\n","protected":false},"author":13,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/344"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/344\/revisions"}],"predecessor-version":[{"id":409,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/344\/revisions\/409"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/344\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=344"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=344"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=344"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}