{"id":338,"date":"2026-02-16T20:52:42","date_gmt":"2026-02-16T20:52:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=338"},"modified":"2026-08-07T19:40:17","modified_gmt":"2026-08-07T19:40:17","slug":"trigonometric-functions-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/trigonometric-functions-get-stronger-answer-key\/","title":{"raw":"Trigonometric Functions: Get Stronger Answer Key","rendered":"Trigonometric Functions: Get Stronger Answer Key"},"content":{"raw":"

Angles<\/h2>\r\n1. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.\r\n\r\n2. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.\r\n\r\n3.\r\n\"Graph\r\n\r\n4.\r\n\"Graph\r\n\r\n5.\r\n\"Graph\r\n\r\n6.\r\n\"Graph\r\n\r\n7.\r\n\"Graph\r\n\r\n8. 240\u00b0\r\n\"Graph<\/span>\r\n\r\n9. [latex]\\frac{4\\pi }{3}[\/latex]\r\n\"Graph<\/span>\r\n\r\n10. [latex]\\frac{2\\pi }{3}[\/latex]\r\n\"Graph<\/span>\r\n\r\n11. [latex]s=\\frac{7\\pi}{3}\\approx 7.33[\/latex]\r\n\r\n12. [latex]\\frac{7\\pi }{2}\\approx 11.00{\\text{ in}}^{2}[\/latex]\r\n\r\n13. [latex]s=\\frac{9\\pi}{5}\\approx 5.65[\/latex]\r\n\r\n14. [latex]\\frac{81\\pi }{20}\\approx 12.72{\\text{ cm}}^{2}[\/latex]\r\n\r\n15. 20\u00b0\r\n\r\n16. 60\u00b0\r\n\r\n17. \u221275\u00b0\r\n\r\n18. [latex]\\frac{\\pi }{2}[\/latex] radians\r\n\r\n19. [latex]-3\\pi [\/latex] radians\r\n\r\n[unmatched: no corresponding problem shown between #35 and #39 in the exercise list \u2014 key answer is \"\u03c0 radians\"]\r\n\r\n20. [latex]\\frac{5\\pi }{6}[\/latex] radians\r\n\r\n21. [latex]\\frac{5.02\\pi }{3}\\approx 5.26[\/latex] miles\r\n\r\n22. [latex]\\frac{21\\pi }{10}\\approx 6.60[\/latex] meters\r\n\r\n23. 104.7198 cm2\r\n\r\n24. 0.7697 in2\r\n\r\n25. 250\u00b0\r\n\r\n26. 320\u00b0\r\n\r\n27. [latex]\\frac{4\\pi }{3}[\/latex]\r\n\r\n28. [latex]\\frac{8\\pi }{9}[\/latex]\r\n\r\n29. 1320 rad 210.085 RPM\r\n\r\n30. 7 in.\/s, 4.77 RPM, 28.65 deg\/s\r\n\r\n31. [latex]5.76[\/latex] miles\r\n\r\n32. [latex]120^\\circ [\/latex]\r\n\r\n33. 11.5 inches\r\n\r\n34. [latex]52^\\circ 22' 30''[\/latex]\r\n\r\n35. [latex]-73^\\circ 15' 0''[\/latex]\r\n\r\n36. [latex]140^\\circ 0' 28.8''[\/latex]\r\n\r\n37. [latex]38.2567^\\circ[\/latex]\r\n\r\n38. [latex]72.0100^\\circ[\/latex]\r\n\r\n39. [latex]-19.5000^\\circ[\/latex]\r\n\r\n40. Complement: [latex]62^\\circ[\/latex], Supplement: [latex]152^\\circ[\/latex]\r\n\r\n41. Complement: [latex]47^\\circ 41' 24''[\/latex], Supplement: [latex]137^\\circ 41' 24''[\/latex]\r\n\r\n42. Complement: No complement exists, Supplement: [latex]68^\\circ[\/latex]\r\n\r\n43. Complement: [latex]\\frac{3\\pi}{8}[\/latex], Supplement: [latex]\\frac{7\\pi}{8}[\/latex]\r\n

Unit Circle: Sine and Cosine Functions<\/h1>\r\n1. On the unit circle, the coordinates of a point corresponding to an angle t are [latex](\\cos t,\\ \\sin t)[\/latex]. The x-coordinate represents [latex]\\cos t[\/latex] and the y-coordinate represents [latex]\\sin t[\/latex].\r\n\r\n2. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.\r\n\r\n3. The sine values are equal.\r\n\r\n4. I\r\n\r\n5. IV\r\n\r\n6. [latex]\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n7. [latex]\\frac{1}{2}[\/latex]\r\n\r\n8. [latex]\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n9. 0\r\n\r\n10. [latex]\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n11. [latex]60^\\circ [\/latex]\r\n\r\n12. [latex]45^\\circ [\/latex]\r\n\r\n13. [latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n14. [latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n15. [latex]60^\\circ [\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]\r\n\r\n16. [latex]45^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n17. [latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n18. [latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]\r\n\r\n19. [latex]\\frac{\\sqrt{77}}{9}[\/latex]\r\n\r\n20. [latex]-\\frac{\\sqrt{15}}{4}[\/latex]\r\n\r\n21. [latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n22. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n23. [latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]\r\n\r\n24. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n25. [latex]\\sin t=0,\\cos t=-1[\/latex]\r\n\r\n26. [latex]\\sin t=\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n27. \u22120.1736\r\n\r\n28. 0.9511\r\n\r\n29. \u22120.1392\r\n\r\n30. \u22120.7660\r\n

Other Trigonometric Functions<\/h1>\r\n1. [latex]\\frac{2\\sqrt{3}}{3}[\/latex]\r\n\r\n2. 1\r\n\r\n3. [latex]-\\frac{2\\sqrt{3}}{3}[\/latex]\r\n\r\n4. [latex]\\sqrt{3}[\/latex]\r\n\r\n5. [latex]\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n6. \u22122\r\n\r\n7. If [latex]\\sin t=-\\frac{2\\sqrt{2}}{3},\\sec t=-3,\\csc t=-\\frac{3\\sqrt{2}}{4},\\tan t=2\\sqrt{2},\\cot t=\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n8. [latex]\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]\r\n\r\n9. [latex]-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n10. 3.1\r\n\r\n11. 1.4\r\n\r\n12. \u20130.228\r\n\r\n13. \u20132.414\r\n\r\n14. 1.414\r\n\r\n15. 1.556\r\n\r\n16. 13.77 hours, period: [latex]1000\\pi [\/latex]\r\n\r\n17. 7.73 inches","rendered":"

Angles<\/h2>\n

1. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.<\/p>\n

2. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.<\/p>\n

3.
\n\"Graph<\/p>\n

4.
\n\"Graph<\/p>\n

5.
\n\"Graph<\/p>\n

6.
\n\"Graph<\/p>\n

7.
\n\"Graph<\/p>\n

8. 240\u00b0
\n\"Graph<\/span><\/p>\n

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

10. [latex]\\frac{2\\pi }{3}[\/latex]
\n\"Graph<\/span><\/p>\n

11. [latex]s=\\frac{7\\pi}{3}\\approx 7.33[\/latex]<\/p>\n

12. [latex]\\frac{7\\pi }{2}\\approx 11.00{\\text{ in}}^{2}[\/latex]<\/p>\n

13. [latex]s=\\frac{9\\pi}{5}\\approx 5.65[\/latex]<\/p>\n

14. [latex]\\frac{81\\pi }{20}\\approx 12.72{\\text{ cm}}^{2}[\/latex]<\/p>\n

15. 20\u00b0<\/p>\n

16. 60\u00b0<\/p>\n

17. \u221275\u00b0<\/p>\n

18. [latex]\\frac{\\pi }{2}[\/latex] radians<\/p>\n

19. [latex]-3\\pi[\/latex] radians<\/p>\n

[unmatched: no corresponding problem shown between #35 and #39 in the exercise list \u2014 key answer is “\u03c0 radians”]<\/p>\n

20. [latex]\\frac{5\\pi }{6}[\/latex] radians<\/p>\n

21. [latex]\\frac{5.02\\pi }{3}\\approx 5.26[\/latex] miles<\/p>\n

22. [latex]\\frac{21\\pi }{10}\\approx 6.60[\/latex] meters<\/p>\n

23. 104.7198 cm2<\/p>\n

24. 0.7697 in2<\/p>\n

25. 250\u00b0<\/p>\n

26. 320\u00b0<\/p>\n

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

28. [latex]\\frac{8\\pi }{9}[\/latex]<\/p>\n

29. 1320 rad 210.085 RPM<\/p>\n

30. 7 in.\/s, 4.77 RPM, 28.65 deg\/s<\/p>\n

31. [latex]5.76[\/latex] miles<\/p>\n

32. [latex]120^\\circ[\/latex]<\/p>\n

33. 11.5 inches<\/p>\n

34. [latex]52^\\circ 22' 30''[\/latex]<\/p>\n

35. [latex]-73^\\circ 15' 0''[\/latex]<\/p>\n

36. [latex]140^\\circ 0' 28.8''[\/latex]<\/p>\n

37. [latex]38.2567^\\circ[\/latex]<\/p>\n

38. [latex]72.0100^\\circ[\/latex]<\/p>\n

39. [latex]-19.5000^\\circ[\/latex]<\/p>\n

40. Complement: [latex]62^\\circ[\/latex], Supplement: [latex]152^\\circ[\/latex]<\/p>\n

41. Complement: [latex]47^\\circ 41' 24''[\/latex], Supplement: [latex]137^\\circ 41' 24''[\/latex]<\/p>\n

42. Complement: No complement exists, Supplement: [latex]68^\\circ[\/latex]<\/p>\n

43. Complement: [latex]\\frac{3\\pi}{8}[\/latex], Supplement: [latex]\\frac{7\\pi}{8}[\/latex]<\/p>\n

Unit Circle: Sine and Cosine Functions<\/h1>\n

1. On the unit circle, the coordinates of a point corresponding to an angle t are [latex](\\cos t,\\ \\sin t)[\/latex]. The x-coordinate represents [latex]\\cos t[\/latex] and the y-coordinate represents [latex]\\sin t[\/latex].<\/p>\n

2. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/p>\n

3. The sine values are equal.<\/p>\n

4. I<\/p>\n

5. IV<\/p>\n

6. [latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

7. [latex]\\frac{1}{2}[\/latex]<\/p>\n

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

9. 0<\/p>\n

10. [latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

11. [latex]60^\\circ[\/latex]<\/p>\n

12. [latex]45^\\circ[\/latex]<\/p>\n

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

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

15. [latex]60^\\circ[\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]<\/p>\n

16. [latex]45^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

17. [latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

18. [latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]<\/p>\n

19. [latex]\\frac{\\sqrt{77}}{9}[\/latex]<\/p>\n

20. [latex]-\\frac{\\sqrt{15}}{4}[\/latex]<\/p>\n

21. [latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

22. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

23. [latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]<\/p>\n

24. [latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n

25. [latex]\\sin t=0,\\cos t=-1[\/latex]<\/p>\n

26. [latex]\\sin t=\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n

27. \u22120.1736<\/p>\n

28. 0.9511<\/p>\n

29. \u22120.1392<\/p>\n

30. \u22120.7660<\/p>\n

Other Trigonometric Functions<\/h1>\n

1. [latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

2. 1<\/p>\n

3. [latex]-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n

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

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

6. \u22122<\/p>\n

7. If [latex]\\sin t=-\\frac{2\\sqrt{2}}{3},\\sec t=-3,\\csc t=-\\frac{3\\sqrt{2}}{4},\\tan t=2\\sqrt{2},\\cot t=\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n

8. [latex]\\sec t=2,\\csc t=\\frac{2\\sqrt{3}}{3},\\tan t=\\sqrt{3},\\cot t=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n

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

10. 3.1<\/p>\n

11. 1.4<\/p>\n

12. \u20130.228<\/p>\n

13. \u20132.414<\/p>\n

14. 1.414<\/p>\n

15. 1.556<\/p>\n

16. 13.77 hours, period: [latex]1000\\pi[\/latex]<\/p>\n

17. 7.73 inches<\/p>\n","protected":false},"author":13,"menu_order":13,"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\/338"}],"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\/338\/revisions"}],"predecessor-version":[{"id":406,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/338\/revisions\/406"}],"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\/338\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=338"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=338"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=338"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}