{"id":2316,"date":"2019-05-10T15:16:04","date_gmt":"2019-05-10T15:16:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2316"},"modified":"2019-05-10T15:19:16","modified_gmt":"2019-05-10T15:19:16","slug":"3-3-section-exercises-2","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/3-3-section-exercises-2\/","title":{"raw":"3.3 Section Exercises","rendered":"3.3 Section Exercises"},"content":{"raw":"

3.3 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. Describe the unit circle.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135453218\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135453218\"]\r\n

The unit circle is a circle of radius 1 centered at the origin.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What do the x-<\/em> and y-<\/em>coordinates of the points on the unit circle represent?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. Discuss the difference between a coterminal angle and a reference angle.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135203835\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135203835\"]\r\n

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,[latex]\\text{ }t,[\/latex] formed by the terminal side of the angle[latex]\\text{ }t\\text{ }[\/latex]and the horizontal axis.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135160175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135160175\"]\r\n

The sine values are equal.<\/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, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by [latex]t[\/latex] lies.<\/p>\r\n\r\n

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6. [latex]\\text{sin}\\left(t\\right)<0\\text{ }[\/latex]and[latex]\\text{ }\\text{cos}\\left(t\\right)<0\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [latex]\\text{sin}\\left(t\\right)>0\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135193433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193433\"]\r\n

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

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8. [latex]\\mathrm{sin}\\left(t\\right)>0[\/latex] and [latex]\\text{ }\\mathrm{cos}\\left(t\\right)<0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]\\mathrm{sin}\\left(t\\right)<0[\/latex] and [latex]\\mathrm{cos}\\left(t\\right)>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135436607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135436607\"]\r\n

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

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

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10. [latex]\\mathrm{sin}\\text{ }\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135499874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135499874\"]\r\n

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

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12. [latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137667848\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137667848\"]\r\n

14. [latex]\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165137823088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137823088\"]\r\n

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

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

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18. [latex]\\mathrm{sin}\\text{ }\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133233941\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133233941\"]\r\n

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

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

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20. [latex]\\mathrm{cos}\\text{ }\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134486775\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134486775\"]\r\n

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

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21. [latex]\\mathrm{cos}\\text{ }0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165133129088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133129088\"]\r\n

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

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[latex]\\mathrm{sin}\\text{ }0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Numeric<\/h4>\r\n

For the following exercises, state the reference angle for the given angle.<\/p>\r\n\r\n

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24. [latex]240\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137817740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137817740\"]\r\n

[latex]60\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]-170\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [latex]100\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135571779\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135571779\"]\r\n

[latex]80\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n27. [latex]-315\u00b0[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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28. [latex]135\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134112806\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134112806\"]\r\n

[latex]45\u00b0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29. [latex]\\frac{5\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]\\frac{2\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165132032593\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132032593\"]\r\n

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

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31. [latex]\\frac{5\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. [latex]\\frac{-11\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133078158\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133078158\"]\r\n

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

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33. [latex]\\frac{-\\text{ }7\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]\\frac{-\\pi }{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137668266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137668266\"]\r\n

[latex]\\frac{\\pi }{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.<\/p>\r\n\r\n

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35. [latex]225\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. [latex]300\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135154333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135154333\"]\r\n

[latex]60\u00b0,[\/latex] Quadrant IV, [latex]\\text{sin}\\left(300\u00b0\\right)=-\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\left(300\u00b0\\right)=\\frac{1}{2}\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. [latex]320\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. [latex]135\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137890678\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137890678\"]\r\n

[latex]45\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(135\u00b0\\right)=\\frac{\\sqrt{2}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(135\u00b0\\right)=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. [latex]210\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [latex]120\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135332783\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135332783\"]\r\n

[latex]60\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(120\u00b0\\right)=\\frac{\\sqrt{3}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(120\u00b0\\right)=-\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. [latex]250\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [latex]150\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165131937940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131937940\"]\r\n

[latex]\\text{ }30\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(150\u00b0\\right)=\\frac{1}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(150\u00b0\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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43. [latex]\\frac{5\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135192734\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135192734\"]\r\n

[latex]\\frac{\\pi }{6},[\/latex] Quadrant III, [latex]\\text{ }\\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165135547159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135547159\"]\r\n

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

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

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48. [latex]\\frac{2\\pi }{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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49. [latex]\\frac{\\pi }{3},[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [latex]\\frac{5\\pi }{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. [latex]\\frac{7\\pi }{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135582120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135582120\"]\r\n

[latex]\\frac{\\pi }{4},[\/latex] Quadrant IV, [latex]\\text{ }\\text{sin}\\left(\\frac{7\\pi }{4}\\right)=-\\frac{\\sqrt{2}}{2},[\/latex][latex]\\text{cos}\\left(\\frac{7\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find the requested value.<\/p>\r\n\r\n

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52. If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{1}{7}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 4th<\/sup> quadrant, find[latex]\\text{ }\\text{sin}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53. If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{2}{9}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 1st<\/sup> quadrant, find[latex]\\text{ }\\text{sin}\\left(t\\right).\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134036846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134036846\"]\r\n

[latex]\\frac{\\sqrt{77}}{9}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54. If[latex]\\text{ }\\text{sin}\\left(t\\right)=\\frac{3}{8}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 2nd<\/sup> quadrant, find[latex]\\text{ }\\text{cos}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. If[latex]\\text{ }\\text{sin}\\left(t\\right)=-\\frac{1}{4}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 3rd<\/sup> quadrant, find[latex]\\text{ }\\text{cos}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135448310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135448310\"]\r\n

[latex]-\\frac{\\sqrt{15}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56. Find the coordinates of the point on a circle with radius 15 corresponding to an angle of[latex]\\text{ }220\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. Find the coordinates of the point on a circle with radius 20 corresponding to an angle of[latex]\\text{ }120\u00b0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133434632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133434632\"]\r\n

[latex]\\text{ }\\left(-10,10\\sqrt{3}\\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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58. Find the coordinates of the point on a circle with radius 8 corresponding to an angle of[latex]\\text{ }\\frac{7\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. Find the coordinates of the point on a circle with radius 16 corresponding to an angle of[latex]\\text{ }\\frac{5\\pi }{9}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137464223\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137464223\"]\r\n

[latex]\\text{ }\\left(\u20132.778,15.757\\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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60. State the domain of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. State the range of the sine and cosine functions.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137854809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137854809\"]\r\n

[latex]\\text{ }\\text{ }\\left[\u20131,1\\right]\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\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 given point on the unit circle to find the value of the sine and cosine of[latex]\\text{ }t\\text{ .}[\/latex]<\/p>\r\n\r\n

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62.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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63.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137453657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137453657\"]\r\n

[latex]\\mathrm{sin}t=\\frac{1}{2},\\mathrm{cos}t=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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65.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137888941\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137888941\"]\r\n

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

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66.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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67.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135195408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135195408\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\text{ }t=-\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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68.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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69.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135251392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135251392\"]\r\n

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

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70.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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71.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135193111\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193111\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=0,\\mathrm{cos}\\text{ }t=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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72.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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73\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137784840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137784840\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=-0.596,\\mathrm{cos}\\text{ }t=0.803[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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74.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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75.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137809935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137809935\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=\\frac{1}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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76.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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77.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137932463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137932463\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=-\\frac{1}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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78.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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79.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137696522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137696522\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=0.761,\\mathrm{cos}\\text{ }t=-0.649[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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80.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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81.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135429391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135429391\"]\r\n

[latex]\\mathrm{sin}\\text{ }t=1,\\mathrm{cos}\\text{ }t=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/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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82. [latex]\\mathrm{sin}\\text{ }\\frac{5\\pi }{9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137415669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137415669\"]\r\n

\u22120.1736<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165137464202\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137464202\"]\r\n

0.9511<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134108607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134108607\"]\r\n

\u22120.7071<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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88. [latex]\\mathrm{sin}\\text{ }98\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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89. [latex]\\mathrm{cos}\\text{ }98\u00b0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135503819\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135503819\"]\r\n

\u22120.1392<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165137400267\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137400267\"]\r\n

\u22120.7660<\/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, evaluate.<\/p>\r\n\r\n

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92. [latex]\\mathrm{sin}\\left(\\frac{11\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{-5\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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93. [latex]\\mathrm{sin}\\left(\\frac{3\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137453874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137453874\"]\r\n

[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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94. [latex]\\mathrm{sin}\\left(-\\frac{4\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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95.[latex]\\mathrm{sin}\\left(\\frac{-9\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{-\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137450898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137450898\"]\r\n

[latex]-\\frac{\\sqrt{6}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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96. [latex]\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\mathrm{cos}\\left(\\frac{-\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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97. [latex]\\mathrm{sin}\\left(\\frac{7\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{-2\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137925605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137925605\"]\r\n

[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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98. [latex]\\mathrm{cos}\\left(\\frac{5\\pi }{6}\\right)\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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99. [latex]\\mathrm{cos}\\left(\\frac{-\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135310629\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135310629\"]\r\n

[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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100. [latex]\\mathrm{sin}\\left(\\frac{-5\\pi }{4}\\right)\\mathrm{sin}\\left(\\frac{11\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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101. [latex]\\mathrm{sin}\\left(\\pi \\right)\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137730448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137730448\"]\r\n

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

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Real-World Applications<\/h4>\r\n

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point[latex]\\text{ }\\left(0,1\\right),[\/latex] that is, on the due north position. Assume the carousel revolves counter clockwise.<\/p>\r\n\r\n

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102. What are the coordinates of the child after 45 seconds?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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103. What are the coordinates of the child after 90 seconds?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135188021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135188021\"]\r\n

[latex]\\left(0,\u20131\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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104. What is the coordinates of the child after 125 seconds?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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105. When will the child have coordinates[latex]\\text{ }\\left(0.707,\u20130.707\\right)\\text{ }[\/latex] if the ride lasts 6 minutes? (There are multiple answers.)<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134085984\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134085984\"]\r\n

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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106. When will the child have coordinates[latex]\\text{ }\\left(-0.866,-0.5\\right)\\text{ }[\/latex] if the ride last 6 minutes?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

3.3 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. Describe the unit circle.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The unit circle is a circle of radius 1 centered at the origin.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2. What do the x-<\/em> and y-<\/em>coordinates of the points on the unit circle represent?<\/p>\n<\/div>\n<\/div>\n

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3. Discuss the difference between a coterminal angle and a reference angle.<\/p>\n<\/div>\n

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

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4. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.<\/p>\n<\/div>\n<\/div>\n

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5. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The sine values are equal.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by [latex]t[\/latex] lies.<\/p>\n

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6. [latex]\\text{sin}\\left(t\\right)<0\\text{ }[\/latex]and[latex]\\text{ }\\text{cos}\\left(t\\right)<0\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [latex]\\text{sin}\\left(t\\right)>0\\text{ }[\/latex]and[latex]\\text{ }\\mathrm{cos}\\left(t\\right)>0[\/latex]<\/p>\n<\/div>\n

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

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8. [latex]\\mathrm{sin}\\left(t\\right)>0[\/latex] and [latex]\\text{ }\\mathrm{cos}\\left(t\\right)<0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]\\mathrm{sin}\\left(t\\right)<0[\/latex] and [latex]\\mathrm{cos}\\left(t\\right)>0[\/latex]<\/p>\n<\/div>\n

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

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

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10. [latex]\\mathrm{sin}\\text{ }\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]\\mathrm{sin}\\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}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [latex]\\mathrm{cos}\\text{ }\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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14. [latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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

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

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18. [latex]\\mathrm{sin}\\text{ }\\pi[\/latex]<\/p>\n<\/div>\n

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

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

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20. [latex]\\mathrm{cos}\\text{ }\\pi[\/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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21. [latex]\\mathrm{cos}\\text{ }0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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

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[latex]\\mathrm{sin}\\text{ }0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Numeric<\/h4>\n

For the following exercises, state the reference angle for the given angle.<\/p>\n

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24. [latex]240\u00b0[\/latex]<\/p>\n<\/div>\n

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[latex]60\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25. [latex]-170\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26. [latex]100\u00b0[\/latex]<\/p>\n<\/div>\n

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[latex]80\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27. [latex]-315\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28. [latex]135\u00b0[\/latex]<\/p>\n<\/div>\n

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[latex]45\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29. [latex]\\frac{5\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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30. [latex]\\frac{2\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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

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

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32. [latex]\\frac{-11\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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

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33. [latex]\\frac{-\\text{ }7\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34. [latex]\\frac{-\\pi }{8}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.<\/p>\n

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35. [latex]225\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36. [latex]300\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]60\u00b0,[\/latex] Quadrant IV, [latex]\\text{sin}\\left(300\u00b0\\right)=-\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\left(300\u00b0\\right)=\\frac{1}{2}\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37. [latex]320\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38. [latex]135\u00b0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]45\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(135\u00b0\\right)=\\frac{\\sqrt{2}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(135\u00b0\\right)=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39. [latex]210\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40. [latex]120\u00b0[\/latex]<\/p>\n<\/div>\n

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[latex]60\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(120\u00b0\\right)=\\frac{\\sqrt{3}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(120\u00b0\\right)=-\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41. [latex]250\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42. [latex]150\u00b0[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ }30\u00b0,[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(150\u00b0\\right)=\\frac{1}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(150\u00b0\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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43. [latex]\\frac{5\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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[latex]\\frac{\\pi }{6},[\/latex] Quadrant III, [latex]\\text{ }\\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<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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

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

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48. [latex]\\frac{2\\pi }{3}[\/latex]<\/p>\n<\/div>\n

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49. [latex]\\frac{\\pi }{3},[\/latex] Quadrant II, [latex]\\text{ }\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2},[\/latex][latex]\\text{ }\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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51. [latex]\\frac{7\\pi }{4}[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\pi }{4},[\/latex] Quadrant IV, [latex]\\text{ }\\text{sin}\\left(\\frac{7\\pi }{4}\\right)=-\\frac{\\sqrt{2}}{2},[\/latex][latex]\\text{cos}\\left(\\frac{7\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find the requested value.<\/p>\n

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52. If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{1}{7}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 4th<\/sup> quadrant, find[latex]\\text{ }\\text{sin}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53. If[latex]\\text{ }\\text{cos}\\left(t\\right)=\\frac{2}{9}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 1st<\/sup> quadrant, find[latex]\\text{ }\\text{sin}\\left(t\\right).\\text{ }[\/latex]<\/p>\n<\/div>\n

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

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54. If[latex]\\text{ }\\text{sin}\\left(t\\right)=\\frac{3}{8}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 2nd<\/sup> quadrant, find[latex]\\text{ }\\text{cos}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. If[latex]\\text{ }\\text{sin}\\left(t\\right)=-\\frac{1}{4}\\text{ }[\/latex]and[latex]\\text{ }t\\text{ }[\/latex]is in the 3rd<\/sup> quadrant, find[latex]\\text{ }\\text{cos}\\left(t\\right).[\/latex]<\/p>\n<\/div>\n

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

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56. Find the coordinates of the point on a circle with radius 15 corresponding to an angle of[latex]\\text{ }220\u00b0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. Find the coordinates of the point on a circle with radius 20 corresponding to an angle of[latex]\\text{ }120\u00b0.[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ }\\left(-10,10\\sqrt{3}\\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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58. Find the coordinates of the point on a circle with radius 8 corresponding to an angle of[latex]\\text{ }\\frac{7\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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59. Find the coordinates of the point on a circle with radius 16 corresponding to an angle of[latex]\\text{ }\\frac{5\\pi }{9}.[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ }\\left(\u20132.778,15.757\\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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60. State the domain of the sine and cosine functions.<\/p>\n<\/div>\n<\/div>\n

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61. State the range of the sine and cosine functions.<\/p>\n<\/div>\n

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[latex]\\text{ }\\text{ }\\left[\u20131,1\\right]\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of[latex]\\text{ }t\\text{ .}[\/latex]<\/p>\n

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62.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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63.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}t=\\frac{1}{2},\\mathrm{cos}t=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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64.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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65.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=-\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\text{ }t=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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66.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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67.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=\\frac{\\sqrt{3}}{2},\\mathrm{cos}\\text{ }t=-\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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68.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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69.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=-\\frac{\\sqrt{2}}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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70.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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71.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=0,\\mathrm{cos}\\text{ }t=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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72.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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73\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=-0.596,\\mathrm{cos}\\text{ }t=0.803[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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74.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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75.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=\\frac{1}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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76.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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77.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=-\\frac{1}{2},\\mathrm{cos}\\text{ }t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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78.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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79.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=0.761,\\mathrm{cos}\\text{ }t=-0.649[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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80.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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81.\u00a0\"Graph<\/span><\/div>\n
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[latex]\\mathrm{sin}\\text{ }t=1,\\mathrm{cos}\\text{ }t=0[\/latex]<\/p>\n<\/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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82. [latex]\\mathrm{sin}\\text{ }\\frac{5\\pi }{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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

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

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

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0.9511<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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\u22120.7071<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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88. [latex]\\mathrm{sin}\\text{ }98\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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89. [latex]\\mathrm{cos}\\text{ }98\u00b0[\/latex]<\/p>\n<\/div>\n

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

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

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

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

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Extensions<\/h4>\n

For the following exercises, evaluate.<\/p>\n

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92. [latex]\\mathrm{sin}\\left(\\frac{11\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{-5\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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93. [latex]\\mathrm{sin}\\left(\\frac{3\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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94. [latex]\\mathrm{sin}\\left(-\\frac{4\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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95.[latex]\\mathrm{sin}\\left(\\frac{-9\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{-\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]-\\frac{\\sqrt{6}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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96. [latex]\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\mathrm{cos}\\left(\\frac{-\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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97. [latex]\\mathrm{sin}\\left(\\frac{7\\pi }{4}\\right)\\mathrm{cos}\\left(\\frac{-2\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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98. [latex]\\mathrm{cos}\\left(\\frac{5\\pi }{6}\\right)\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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99. [latex]\\mathrm{cos}\\left(\\frac{-\\pi }{3}\\right)\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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100. [latex]\\mathrm{sin}\\left(\\frac{-5\\pi }{4}\\right)\\mathrm{sin}\\left(\\frac{11\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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101. [latex]\\mathrm{sin}\\left(\\pi \\right)\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n

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

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Real-World Applications<\/h4>\n

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point[latex]\\text{ }\\left(0,1\\right),[\/latex] that is, on the due north position. Assume the carousel revolves counter clockwise.<\/p>\n

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102. What are the coordinates of the child after 45 seconds?<\/p>\n<\/div>\n<\/div>\n

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103. What are the coordinates of the child after 90 seconds?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\left(0,\u20131\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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104. What is the coordinates of the child after 125 seconds?<\/p>\n<\/div>\n<\/div>\n

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105. When will the child have coordinates[latex]\\text{ }\\left(0.707,\u20130.707\\right)\\text{ }[\/latex] if the ride lasts 6 minutes? (There are multiple answers.)<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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106. When will the child have coordinates[latex]\\text{ }\\left(-0.866,-0.5\\right)\\text{ }[\/latex] if the ride last 6 minutes?<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158103,"menu_order":6,"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-2316","chapter","type-chapter","status-web-only","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2316","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\/2316\/revisions"}],"predecessor-version":[{"id":2320,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2316\/revisions\/2320"}],"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\/2316\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2316"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2316"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2316"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2316"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}