{"id":2333,"date":"2019-05-12T21:15:28","date_gmt":"2019-05-12T21:15:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2333"},"modified":"2019-05-15T16:54:26","modified_gmt":"2019-05-15T16:54:26","slug":"3-5-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/3-5-section-exercises\/","title":{"raw":"3.5 Section Exercises","rendered":"3.5 Section Exercises"},"content":{"raw":"
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3.5 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. On an interval of[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex] can the sine and cosine values of a radian measure ever be equal? If so, where?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135419854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135419854\"]\r\n

Yes, when the reference angle is[latex]\\text{ }\\frac{\\pi }{4}\\text{ }[\/latex]and the terminal side of the angle is in quadrants I and III. Thus, at[latex]\\text{ }x=\\frac{\\pi }{4},\\frac{5\\pi }{4},[\/latex] the sine and cosine values are equal.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What would you estimate the cosine of[latex]\\text{ }\\pi \\text{ }[\/latex]degrees to be? Explain your reasoning.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137443119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137443119\"]\r\n

Substitute the sine of the angle in for[latex]\\text{ }y\\text{ }[\/latex]in the Pythagorean Theorem[latex]\\text{ }{x}^{2}+{y}^{2}=1.\\text{ }[\/latex]Solve for[latex]\\text{ }x\\text{ }[\/latex]and take the negative solution.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Describe the secant function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. Tangent and cotangent have a period of[latex]\\text{ }\\pi .\\text{ }[\/latex] What does this tell us about the output of these functions?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135531630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135531630\"]\r\n

The outputs of tangent and cotangent will repeat every[latex]\\text{ }\\pi \\text{ }[\/latex] units.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n

For the following exercises, find the exact value of each expression.<\/p>\r\n\r\n

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6. [latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137466453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137466453\"]\r\n

[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137679375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137679375\"]\r\n

[latex]\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137638454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137638454\"]\r\n

[latex]\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0 [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135407195\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135407195\"]\r\n

1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134239747\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134239747\"]\r\n

2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134357592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134357592\"]\r\n

[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use reference angles to evaluate the expression.<\/p>\r\n\r\n

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18. [latex]\\mathrm{tan}\\text{ }\\frac{5\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]\\mathrm{sec}\\text{ }\\frac{7\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135191609\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191609\"]\r\n

[latex]-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]\\mathrm{csc}\\text{ }\\frac{11\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]\\mathrm{cot}\\text{ }\\frac{13\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134417902\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134417902\"][latex]\\sqrt{3}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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[latex]\\mathrm{tan}\\text{ }\\frac{7\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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[latex]\\mathrm{sec}\\text{ }\\frac{3\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137552229\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137552229\"]\r\n

[latex]-\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]\\mathrm{csc}\\text{ }\\frac{5\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]\\mathrm{cot}\\text{ }\\frac{11\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137732040\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137732040\"]\r\n

\u22121<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [latex]\\mathrm{tan}\\text{ }\\frac{8\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]\\mathrm{sec}\\text{ }\\frac{4\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137430538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137430538\"]\r\n

\u22122<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [latex]\\mathrm{csc}\\text{ }\\frac{2\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]\\mathrm{cot}\\text{ }\\frac{5\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135440207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135440207\"]\r\n

[latex]-\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n28. [latex]\\mathrm{tan}\\text{ }225\u00b0[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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29. [latex]\\mathrm{sec}\\text{ }300\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137583322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137583322\"]\r\n

2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]\\mathrm{csc}\\text{ }150\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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31. [latex]\\mathrm{cot}\\text{ }240\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137762570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137762570\"]\r\n

[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. [latex]\\mathrm{tan}\\text{ }330\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. [latex]\\mathrm{sec}\\text{ }120\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137405618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137405618\"]\r\n

\u22122<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]\\mathrm{csc}\\text{ }210\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]\\mathrm{cot}\\text{ }315\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137600949\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137600949\"]\r\n

\u22121<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. If[latex]\\text{ }\\text{sin}\\text{ }t=\\frac{3}{4},[\/latex] and[latex]\\text{ }\\text{ }t\\text{ }[\/latex]is in quadrant II, find[latex]\\text{ }\\mathrm{cos}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. If[latex]\\text{ }\\text{cos}\\text{ }t=-\\frac{1}{3},[\/latex] and[latex]\\text{ }\\text{ }t\\text{ }[\/latex]is in quadrant III, find[latex]\\text{ }\\mathrm{sin}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134381672\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134381672\"]\r\n

If[latex]\\mathrm{sin}\\text{ }t=-\\frac{2\\sqrt{2}}{3},\\mathrm{sec}\\text{ }t=-3,\\mathrm{csc}\\text{ }t=-\\frac{3\\sqrt{2}}{4},\\mathrm{tan}\\text{ }t=2\\sqrt{2},\\mathrm{cot}\\text{ }t=\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. If[latex]\\text{ }\\mathrm{tan}\\text{ }t=\\frac{12}{5},[\/latex] and[latex]\\text{ }0\\le t<\\frac{\\pi }{2},[\/latex] find[latex]\\text{ }\\mathrm{sin}\\text{ }t,\\mathrm{cos}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39..If[latex]\\text{ }\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{3}}{2}\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\text{ }t=\\frac{1}{2},[\/latex] find[latex]\\text{ }\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[latex]\\mathrm{sec}\\text{ }t=2,\\mathrm{csc}\\text{ }t=\\frac{2\\sqrt{3}}{3},\\text{ }\\mathrm{tan}\\text{ }t=\\sqrt{3},\\text{ }\\mathrm{cot}\\text{ }t=\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. If[latex]\\text{ }\\mathrm{sin}\\text{ }40\u00b0\\approx 0.643\\text{ }\\text{ }\\mathrm{cos}\\text{ }40\u00b0\\approx 0.766\\text{ }\\text{ }\\text{sec}\\text{ }40\u00b0,\\text{csc}\\text{ }40\u00b0,\\text{tan}\\text{ }40\u00b0,\\text{and}\\text{ }\\text{cot}\\text{ }40\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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If[latex]\\text{ }\\text{sin}\\text{ }t=\\frac{\\sqrt{2}}{2},[\/latex] what is the[latex]\\text{ }\\text{sin}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137777664\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137777664\"]\r\n

[latex]\\text{ }-\\frac{\\sqrt{2}}{2}\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. If[latex]\\text{ }\\text{cos}\\text{ }t=\\frac{1}{2},[\/latex] what is the[latex]\\text{ }\\text{cos}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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If[latex]\\text{ }\\text{sec}\\text{ }t=3.1,[\/latex] what is the[latex]\\text{ }\\text{sec}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135388457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135388457\"]\r\n

3.1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. If[latex]\\text{ }\\text{csc}\\text{ }t=0.34,[\/latex] what is the[latex]\\text{ }\\text{csc}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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43. If[latex]\\text{ }\\text{tan}\\text{ }t=-1.4,[\/latex] what is the[latex]\\text{ }\\text{tan}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137437845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137437845\"]\r\n

1.4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44. If[latex]\\text{ }\\text{cot}\\text{ }t=9.23,[\/latex] what is the[latex]\\text{ }\\text{cot}\\left(-t\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.<\/p>\r\n\r\n

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45.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137942478\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137942478\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{2}}{2},\\mathrm{tan}\\text{ }t=1,\\mathrm{cot}\\text{ }t=1,\\mathrm{sec}\\text{ }t=\\sqrt{2},\\mathrm{csc}\\text{ }t=\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0\"Graph<\/div>\r\n<\/div>\r\n
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47.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134398694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134398694\"][latex]\\mathrm{sin}\\text{ }t=-\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\text{ }t=-\\frac{1}{2},\\mathrm{tan}\\text{ }t=\\sqrt{3},\\mathrm{cot}\\text{ }t=\\frac{\\sqrt{3}}{3},\\mathrm{sec}\\text{ }t=-2,\\mathrm{csc}\\text{ }t=-\\frac{2\\sqrt{3}}{3}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Technology<\/h4>\r\n

For the following exercises, use a graphing calculator to evaluate.<\/p>\r\n\r\n

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48. [latex]\\mathrm{csc}\\text{ }\\frac{5\\pi }{9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]\\mathrm{cot}\\text{ }\\frac{4\\pi }{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134152620\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134152620\"]\r\n

\u20130.228<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{10}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. [latex]\\mathrm{tan}\\text{ }\\frac{5\\pi }{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135548933\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135548933\"]\r\n

\u20132.414<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52. [latex]\\mathrm{sec}\\text{ }\\frac{3\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135255105\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135255105\"]\r\n

1.414<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54. [latex]\\text{tan}\\text{ }98\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. [latex]\\mathrm{cot}\\text{ }33\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137461234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137461234\"]\r\n

1.540<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56. [latex]\\mathrm{cot}\\text{ }140\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]\\mathrm{sec}\\text{ }310\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134328212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134328212\"]\r\n

1.556<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n

For the following exercises, use identities to evaluate the expression.<\/p>\r\n\r\n

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58. If[latex]\\text{ }\\mathrm{tan}\\left(t\\right)\\approx 2.7,[\/latex] and[latex]\\text{ }\\mathrm{sin}\\left(t\\right)\\approx 0.94,[\/latex] find[latex]\\text{ }\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n59. If[latex]\\text{ }\\mathrm{tan}\\left(t\\right)\\approx 1.3,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.61,[\/latex] find[latex]\\text{ }\\mathrm{sin}\\left(t\\right).\\text{ }[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137646930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137646930\"]\r\n

[latex]\\mathrm{sin}\\left(t\\right)\\approx 0.79[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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60. If[latex]\\text{ }\\mathrm{csc}\\left(t\\right)\\approx 3.2,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.95,[\/latex] find[latex]\\text{ }\\mathrm{tan}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. If[latex]\\text{ }\\mathrm{cot}\\left(t\\right)\\approx 0.58,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.5,[\/latex] find[latex]\\text{ }\\mathrm{csc}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137442249\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137442249\"]\r\n

[latex]\\mathrm{csc}t\\approx 1.16[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. Determine whether the function[latex]\\text{ }f\\left(x\\right)=2\\mathrm{sin}\\text{ }x\\text{ }\\mathrm{cos}\\text{ }x\\text{ }[\/latex]is even, odd, or neither.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. Determine whether the function [latex]f\\left(x\\right)=3{\\mathrm{sin}}^{2}x\\text{ }\\mathrm{cos}\\text{ }x\\text{ }+\\text{ }\\mathrm{sec}\\text{ }x[\/latex] is even, odd, or neither.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135453321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135453321\"]\r\n

even<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. Determine whether the function [latex]f\\left(x\\right)=\\mathrm{sin}\\text{ }x\\text{ }-2{\\mathrm{cos}}^{2}x[\/latex] is even, odd, or neither.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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65. Determine whether the function[latex]\\text{ }f\\left(x\\right)={\\mathrm{csc}}^{2}x+\\mathrm{sec}\\text{ }x\\text{ }[\/latex]is even, odd, or neither.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137604862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137604862\"]\r\n

even<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use identities to simplify the expression.<\/p>\r\n\r\n

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66. [latex]\\mathrm{csc}\\text{ }t\\text{ }\\mathrm{tan}\\text{ }t[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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67. [latex]\\frac{\\mathrm{sec}\\text{ }t}{\\mathrm{csc}\\text{ }t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134566552\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134566552\"][latex]\\frac{\\mathrm{sin}\\text{ }t}{\\mathrm{cos}\\text{ }t}=\\mathrm{tan}\\text{ }t[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Real-World Applications<\/h4>\r\n
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68. The amount of sunlight in a certain city can be modeled by the function[latex]\\text{ }h=15\\mathrm{cos}\\left(\\frac{1}{600}d\\right),[\/latex] where[latex]\\text{ }h\\text{ }[\/latex]represents the hours of sunlight, and[latex]\\text{ }d\\text{ }[\/latex]is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42nd<\/sup> day of the year. State the period of the function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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69. The amount of sunlight in a certain city can be modeled by the function [latex]\\text{ }h=16\\mathrm{cos}\\left(\\frac{1}{500}d\\right),[\/latex] where[latex]\\text{ }h\\text{ }[\/latex] represents the hours of sunlight, and[latex]\\text{ }d\\text{ }[\/latex] is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th<\/sup> day of the year. State the period of the function.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135344897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135344897\"]\r\n

13.77 hours, period:[latex]\\text{ }1000\\pi \\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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70. The equation[latex]\\text{ }P=20\\mathrm{sin}\\left(2\\pi t\\right)+100\\text{ }[\/latex]models the blood pressure,[latex]\\text{ }P,[\/latex] where[latex]\\text{ }t\\text{ }[\/latex]represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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71. The height of a piston,[latex]\\text{ }h,[\/latex] in inches, can be modeled by the equation[latex]\\text{ }y=2\\mathrm{cos}\\text{ }x+6,[\/latex] where[latex]\\text{ }x\\text{ }[\/latex]represents the crank angle. Find the height of the piston when the crank angle is[latex]\\text{ }55\u00b0.\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134346370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134346370\"]\r\n

7.73 inches<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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72. The height of a piston,[latex]\\text{ }h,[\/latex]in inches, can be modeled by the equation[latex]\\text{ }y=2\\mathrm{cos}\\text{ }x+5,[\/latex]where[latex]\\text{ }x\\text{ }[\/latex]represents the crank angle. Find the height of the piston when the crank angle is [latex]\\text{ }55\u00b0.\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

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3.5 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. On an interval of[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex] can the sine and cosine values of a radian measure ever be equal? If so, where?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

Yes, when the reference angle is[latex]\\text{ }\\frac{\\pi }{4}\\text{ }[\/latex]and the terminal side of the angle is in quadrants I and III. Thus, at[latex]\\text{ }x=\\frac{\\pi }{4},\\frac{5\\pi }{4},[\/latex] the sine and cosine values are equal.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2. What would you estimate the cosine of[latex]\\text{ }\\pi \\text{ }[\/latex]degrees to be? Explain your reasoning.<\/p>\n<\/div>\n<\/div>\n

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3. For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

Substitute the sine of the angle in for[latex]\\text{ }y\\text{ }[\/latex]in the Pythagorean Theorem[latex]\\text{ }{x}^{2}+{y}^{2}=1.\\text{ }[\/latex]Solve for[latex]\\text{ }x\\text{ }[\/latex]and take the negative solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4. Describe the secant function.<\/p>\n<\/div>\n<\/div>\n

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5. Tangent and cotangent have a period of[latex]\\text{ }\\pi .\\text{ }[\/latex] What does this tell us about the output of these functions?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The outputs of tangent and cotangent will repeat every[latex]\\text{ }\\pi \\text{ }[\/latex] units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, find the exact value of each expression.<\/p>\n

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6. [latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [latex]\\mathrm{tan}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12.\u00a0 [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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15. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [latex]\\mathrm{cot}\\text{ }\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use reference angles to evaluate the expression.<\/p>\n

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18. [latex]\\mathrm{tan}\\text{ }\\frac{5\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]\\mathrm{sec}\\text{ }\\frac{7\\pi }{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]-\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20. [latex]\\mathrm{csc}\\text{ }\\frac{11\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]\\mathrm{cot}\\text{ }\\frac{13\\pi }{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\sqrt{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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[latex]\\mathrm{tan}\\text{ }\\frac{7\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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[latex]\\mathrm{sec}\\text{ }\\frac{3\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]-\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22. [latex]\\mathrm{csc}\\text{ }\\frac{5\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]\\mathrm{cot}\\text{ }\\frac{11\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u22121<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24. [latex]\\mathrm{tan}\\text{ }\\frac{8\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25. [latex]\\mathrm{sec}\\text{ }\\frac{4\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u22122<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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26. [latex]\\mathrm{csc}\\text{ }\\frac{2\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27. [latex]\\mathrm{cot}\\text{ }\\frac{5\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]-\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28. [latex]\\mathrm{tan}\\text{ }225\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29. [latex]\\mathrm{sec}\\text{ }300\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. [latex]\\mathrm{csc}\\text{ }150\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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31. [latex]\\mathrm{cot}\\text{ }240\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32. [latex]\\mathrm{tan}\\text{ }330\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33. [latex]\\mathrm{sec}\\text{ }120\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u22122<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34. [latex]\\mathrm{csc}\\text{ }210\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35. [latex]\\mathrm{cot}\\text{ }315\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u22121<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36. If[latex]\\text{ }\\text{sin}\\text{ }t=\\frac{3}{4},[\/latex] and[latex]\\text{ }\\text{ }t\\text{ }[\/latex]is in quadrant II, find[latex]\\text{ }\\mathrm{cos}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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37. If[latex]\\text{ }\\text{cos}\\text{ }t=-\\frac{1}{3},[\/latex] and[latex]\\text{ }\\text{ }t\\text{ }[\/latex]is in quadrant III, find[latex]\\text{ }\\mathrm{sin}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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If[latex]\\mathrm{sin}\\text{ }t=-\\frac{2\\sqrt{2}}{3},\\mathrm{sec}\\text{ }t=-3,\\mathrm{csc}\\text{ }t=-\\frac{3\\sqrt{2}}{4},\\mathrm{tan}\\text{ }t=2\\sqrt{2},\\mathrm{cot}\\text{ }t=\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38. If[latex]\\text{ }\\mathrm{tan}\\text{ }t=\\frac{12}{5},[\/latex] and[latex]\\text{ }0\\le t<\\frac{\\pi }{2},[\/latex] find[latex]\\text{ }\\mathrm{sin}\\text{ }t,\\mathrm{cos}\\text{ }t,\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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39..If[latex]\\text{ }\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{3}}{2}\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\text{ }t=\\frac{1}{2},[\/latex] find[latex]\\text{ }\\mathrm{sec}\\text{ }t,\\mathrm{csc}\\text{ }t,\\mathrm{tan}\\text{ }t,[\/latex] and [latex]\\text{ }\\mathrm{cot}\\text{ }t.[\/latex]<\/p>\n<\/div>\n

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

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40. If[latex]\\text{ }\\mathrm{sin}\\text{ }40\u00b0\\approx 0.643\\text{ }\\text{ }\\mathrm{cos}\\text{ }40\u00b0\\approx 0.766\\text{ }\\text{ }\\text{sec}\\text{ }40\u00b0,\\text{csc}\\text{ }40\u00b0,\\text{tan}\\text{ }40\u00b0,\\text{and}\\text{ }\\text{cot}\\text{ }40\u00b0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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If[latex]\\text{ }\\text{sin}\\text{ }t=\\frac{\\sqrt{2}}{2},[\/latex] what is the[latex]\\text{ }\\text{sin}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{ }-\\frac{\\sqrt{2}}{2}\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41. If[latex]\\text{ }\\text{cos}\\text{ }t=\\frac{1}{2},[\/latex] what is the[latex]\\text{ }\\text{cos}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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If[latex]\\text{ }\\text{sec}\\text{ }t=3.1,[\/latex] what is the[latex]\\text{ }\\text{sec}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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3.1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42. If[latex]\\text{ }\\text{csc}\\text{ }t=0.34,[\/latex] what is the[latex]\\text{ }\\text{csc}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43. If[latex]\\text{ }\\text{tan}\\text{ }t=-1.4,[\/latex] what is the[latex]\\text{ }\\text{tan}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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1.4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44. If[latex]\\text{ }\\text{cot}\\text{ }t=9.23,[\/latex] what is the[latex]\\text{ }\\text{cot}\\left(-t\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.<\/p>\n

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45.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{2}}{2},\\mathrm{tan}\\text{ }t=1,\\mathrm{cot}\\text{ }t=1,\\mathrm{sec}\\text{ }t=\\sqrt{2},\\mathrm{csc}\\text{ }t=\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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46.\u00a0\"Graph<\/div>\n<\/div>\n
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47.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
[latex]\\mathrm{sin}\\text{ }t=-\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\text{ }t=-\\frac{1}{2},\\mathrm{tan}\\text{ }t=\\sqrt{3},\\mathrm{cot}\\text{ }t=\\frac{\\sqrt{3}}{3},\\mathrm{sec}\\text{ }t=-2,\\mathrm{csc}\\text{ }t=-\\frac{2\\sqrt{3}}{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Technology<\/h4>\n

For the following exercises, use a graphing calculator to evaluate.<\/p>\n

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48. [latex]\\mathrm{csc}\\text{ }\\frac{5\\pi }{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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49. [latex]\\mathrm{cot}\\text{ }\\frac{4\\pi }{7}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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\u20130.228<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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50. [latex]\\mathrm{sec}\\text{ }\\frac{\\pi }{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51. [latex]\\mathrm{tan}\\text{ }\\frac{5\\pi }{8}[\/latex]<\/p>\n<\/div>\n

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

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52. [latex]\\mathrm{sec}\\text{ }\\frac{3\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53. [latex]\\mathrm{csc}\\text{ }\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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1.414<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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54. [latex]\\text{tan}\\text{ }98\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. [latex]\\mathrm{cot}\\text{ }33\u00b0[\/latex]<\/p>\n<\/div>\n

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1.540<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56. [latex]\\mathrm{cot}\\text{ }140\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. [latex]\\mathrm{sec}\\text{ }310\u00b0[\/latex]<\/p>\n<\/div>\n

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1.556<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n

For the following exercises, use identities to evaluate the expression.<\/p>\n

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58. If[latex]\\text{ }\\mathrm{tan}\\left(t\\right)\\approx 2.7,[\/latex] and[latex]\\text{ }\\mathrm{sin}\\left(t\\right)\\approx 0.94,[\/latex] find[latex]\\text{ }\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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59. If[latex]\\text{ }\\mathrm{tan}\\left(t\\right)\\approx 1.3,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.61,[\/latex] find[latex]\\text{ }\\mathrm{sin}\\left(t\\right).\\text{ }[\/latex]<\/p>\n<\/div>\n

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[latex]\\mathrm{sin}\\left(t\\right)\\approx 0.79[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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60. If[latex]\\text{ }\\mathrm{csc}\\left(t\\right)\\approx 3.2,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.95,[\/latex] find[latex]\\text{ }\\mathrm{tan}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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61. If[latex]\\text{ }\\mathrm{cot}\\left(t\\right)\\approx 0.58,[\/latex] and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)\\approx 0.5,[\/latex] find[latex]\\text{ }\\mathrm{csc}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n

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[latex]\\mathrm{csc}t\\approx 1.16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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62. Determine whether the function[latex]\\text{ }f\\left(x\\right)=2\\mathrm{sin}\\text{ }x\\text{ }\\mathrm{cos}\\text{ }x\\text{ }[\/latex]is even, odd, or neither.<\/p>\n<\/div>\n<\/div>\n

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63. Determine whether the function [latex]f\\left(x\\right)=3{\\mathrm{sin}}^{2}x\\text{ }\\mathrm{cos}\\text{ }x\\text{ }+\\text{ }\\mathrm{sec}\\text{ }x[\/latex] is even, odd, or neither.<\/p>\n<\/div>\n

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even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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64. Determine whether the function [latex]f\\left(x\\right)=\\mathrm{sin}\\text{ }x\\text{ }-2{\\mathrm{cos}}^{2}x[\/latex] is even, odd, or neither.<\/p>\n<\/div>\n<\/div>\n

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65. Determine whether the function[latex]\\text{ }f\\left(x\\right)={\\mathrm{csc}}^{2}x+\\mathrm{sec}\\text{ }x\\text{ }[\/latex]is even, odd, or neither.<\/p>\n<\/div>\n

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even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use identities to simplify the expression.<\/p>\n

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66. [latex]\\mathrm{csc}\\text{ }t\\text{ }\\mathrm{tan}\\text{ }t[\/latex]<\/p>\n<\/div>\n<\/div>\n

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67. [latex]\\frac{\\mathrm{sec}\\text{ }t}{\\mathrm{csc}\\text{ }t}[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\mathrm{sin}\\text{ }t}{\\mathrm{cos}\\text{ }t}=\\mathrm{tan}\\text{ }t[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Real-World Applications<\/h4>\n
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68. The amount of sunlight in a certain city can be modeled by the function[latex]\\text{ }h=15\\mathrm{cos}\\left(\\frac{1}{600}d\\right),[\/latex] where[latex]\\text{ }h\\text{ }[\/latex]represents the hours of sunlight, and[latex]\\text{ }d\\text{ }[\/latex]is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42nd<\/sup> day of the year. State the period of the function.<\/p>\n<\/div>\n<\/div>\n

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69. The amount of sunlight in a certain city can be modeled by the function [latex]\\text{ }h=16\\mathrm{cos}\\left(\\frac{1}{500}d\\right),[\/latex] where[latex]\\text{ }h\\text{ }[\/latex] represents the hours of sunlight, and[latex]\\text{ }d\\text{ }[\/latex] is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th<\/sup> day of the year. State the period of the function.<\/p>\n<\/div>\n

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13.77 hours, period:[latex]\\text{ }1000\\pi \\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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70. The equation[latex]\\text{ }P=20\\mathrm{sin}\\left(2\\pi t\\right)+100\\text{ }[\/latex]models the blood pressure,[latex]\\text{ }P,[\/latex] where[latex]\\text{ }t\\text{ }[\/latex]represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?<\/p>\n<\/div>\n<\/div>\n

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71. The height of a piston,[latex]\\text{ }h,[\/latex] in inches, can be modeled by the equation[latex]\\text{ }y=2\\mathrm{cos}\\text{ }x+6,[\/latex] where[latex]\\text{ }x\\text{ }[\/latex]represents the crank angle. Find the height of the piston when the crank angle is[latex]\\text{ }55\u00b0.\\text{ }[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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7.73 inches<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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72. The height of a piston,[latex]\\text{ }h,[\/latex]in inches, can be modeled by the equation[latex]\\text{ }y=2\\mathrm{cos}\\text{ }x+5,[\/latex]where[latex]\\text{ }x\\text{ }[\/latex]represents the crank angle. Find the height of the piston when the crank angle is [latex]\\text{ }55\u00b0.\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158103,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2333","chapter","type-chapter","status-web-only","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2333","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158103"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2333\/revisions"}],"predecessor-version":[{"id":2342,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2333\/revisions\/2342"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2333\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2333"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2333"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2333"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}