{"id":13814,"date":"2018-06-14T23:51:29","date_gmt":"2018-06-14T23:51:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/solutions-12\/"},"modified":"2021-11-22T21:49:06","modified_gmt":"2021-11-22T21:49:06","slug":"solutions-unit-circle","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solutions-unit-circle\/","title":{"raw":"Solutions: Unit Circle","rendered":"Solutions: Unit Circle"},"content":{"raw":"<h2>Solutions to Try Its<\/h2>\r\n1.\u00a0[latex]\\cos \\left(t\\right)=-\\frac{\\sqrt{2}}{2},\\sin \\left(t\\right)=\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n2.\u00a0[latex]\\cos \\left(\\pi \\right)=-1[\/latex], [latex]\\sin \\left(\\pi \\right)=0[\/latex]\r\n\r\n3.\u00a0[latex]\\sin \\left(t\\right)=-\\frac{7}{25}[\/latex]\r\n\r\n4.\u00a0approximately 0.866025403\r\n\r\n5.\u00a0[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n6.\u00a0a. [latex]\\text{cos}\\left(315^\\circ \\right)=\\frac{\\sqrt{2}}{2},\\text{sin}\\left(315^\\circ \\right)=\\frac{-\\sqrt{2}}{2}[\/latex]\r\nb. [latex]\\cos \\left(-\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2},\\sin \\left(-\\frac{\\pi }{6}\\right)=-\\frac{1}{2}[\/latex]\r\n\r\n7.\u00a0[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]\r\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\r\n1. The unit circle is a circle of radius 1 centered at the origin.\r\n\r\n3.\u00a0Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.\r\n\r\n5.\u00a0The sine values are equal.\r\n\r\n7. I\r\n\r\n9. IV\r\n\r\n11.\u00a0[latex]\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{1}{2}[\/latex]\r\n\r\n15.\u00a0[latex]\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n17. 0\r\n\r\n19.\u00a0\u22121\r\n\r\n21.\u00a0[latex]\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n23.\u00a0[latex]60^\\circ [\/latex]\r\n\r\n25.\u00a0[latex]80^\\circ [\/latex]\r\n\r\n27.\u00a0[latex]45^\\circ [\/latex]\r\n\r\n29.\u00a0[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{\\pi }{3}[\/latex]\r\n\r\n33.\u00a0[latex]\\frac{\\pi }{8}[\/latex]\r\n\r\n35.\u00a0[latex]60^\\circ [\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]\r\n\r\n37.\u00a0[latex]45^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n39.\u00a0[latex]60^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(120^\\circ \\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(120^\\circ \\right)=-\\frac{1}{2}[\/latex]\r\n\r\n41.\u00a0[latex]30^\\circ [\/latex], Quadrant II, [latex]\\text{sin}\\left(150^\\circ \\right)=\\frac{1}{2}[\/latex], [latex]\\cos \\left(150^\\circ \\right)=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n43.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n45.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]\r\n\r\n47.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]\r\n\r\n49.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\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]\r\n\r\n51.\u00a0[latex]\\frac{\\sqrt{77}}{9}[\/latex]\r\n\r\n53.\u00a0[latex]-\\frac{\\sqrt{15}}{4}[\/latex]\r\n\r\n55.\u00a0[latex]\\left(-10,10\\sqrt{3}\\right)[\/latex]\r\n\r\n57.\u00a0[latex]\\left(-2.778,15.757\\right)[\/latex]\r\n\r\n59.\u00a0[latex]\\left[-1,1\\right][\/latex]\r\n\r\n61.\u00a0[latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n63.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n65.\u00a0[latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]\r\n\r\n67.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n69.\u00a0[latex]\\sin t=0,\\cos t=-1[\/latex]\r\n\r\n71.\u00a0[latex]\\sin t=-0.596,\\cos t=0.803[\/latex]\r\n\r\n73.\u00a0[latex]\\sin t=\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n75.\u00a0[latex]\\sin t=-\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]\r\n\r\n77.\u00a0[latex]\\sin t=0.761,\\cos t=-0.649[\/latex]\r\n\r\n79.\u00a0[latex]\\sin t=1,\\cos t=0[\/latex]\r\n\r\n81.\u00a0\u22120.1736\r\n\r\n83.\u00a00.9511\r\n\r\n85.\u00a0\u22120.7071\r\n\r\n87.\u00a0\u22120.1392\r\n\r\n89.\u00a0\u22120.7660\r\n\r\n91.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n93.\u00a0[latex]-\\frac{\\sqrt{6}}{4}[\/latex]\r\n\r\n95.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n97.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]\r\n\r\n99. 0\r\n\r\n101.\u00a0[latex]\\left(0,-1\\right)[\/latex]\r\n\r\n103.\u00a037.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds","rendered":"<h2>Solutions to Try Its<\/h2>\n<p>1.\u00a0[latex]\\cos \\left(t\\right)=-\\frac{\\sqrt{2}}{2},\\sin \\left(t\\right)=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p>2.\u00a0[latex]\\cos \\left(\\pi \\right)=-1[\/latex], [latex]\\sin \\left(\\pi \\right)=0[\/latex]<\/p>\n<p>3.\u00a0[latex]\\sin \\left(t\\right)=-\\frac{7}{25}[\/latex]<\/p>\n<p>4.\u00a0approximately 0.866025403<\/p>\n<p>5.\u00a0[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<p>6.\u00a0a. [latex]\\text{cos}\\left(315^\\circ \\right)=\\frac{\\sqrt{2}}{2},\\text{sin}\\left(315^\\circ \\right)=\\frac{-\\sqrt{2}}{2}[\/latex]<br \/>\nb. [latex]\\cos \\left(-\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2},\\sin \\left(-\\frac{\\pi }{6}\\right)=-\\frac{1}{2}[\/latex]<\/p>\n<p>7.\u00a0[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1. The unit circle is a circle of radius 1 centered at the origin.<\/p>\n<p>3.\u00a0Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis.<\/p>\n<p>5.\u00a0The sine values are equal.<\/p>\n<p>7. I<\/p>\n<p>9. IV<\/p>\n<p>11.\u00a0[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>13.\u00a0[latex]\\frac{1}{2}[\/latex]<\/p>\n<p>15.\u00a0[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p>17. 0<\/p>\n<p>19.\u00a0\u22121<\/p>\n<p>21.\u00a0[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>23.\u00a0[latex]60^\\circ[\/latex]<\/p>\n<p>25.\u00a0[latex]80^\\circ[\/latex]<\/p>\n<p>27.\u00a0[latex]45^\\circ[\/latex]<\/p>\n<p>29.\u00a0[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<p>31.\u00a0[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<p>33.\u00a0[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n<p>35.\u00a0[latex]60^\\circ[\/latex], Quadrant IV, [latex]\\text{sin}\\left(300^\\circ \\right)=-\\frac{\\sqrt{3}}{2},\\cos \\left(300^\\circ \\right)=\\frac{1}{2}[\/latex]<\/p>\n<p>37.\u00a0[latex]45^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(135^\\circ \\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(135^\\circ \\right)=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p>39.\u00a0[latex]60^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(120^\\circ \\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(120^\\circ \\right)=-\\frac{1}{2}[\/latex]<\/p>\n<p>41.\u00a0[latex]30^\\circ[\/latex], Quadrant II, [latex]\\text{sin}\\left(150^\\circ \\right)=\\frac{1}{2}[\/latex], [latex]\\cos \\left(150^\\circ \\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>43.\u00a0[latex]\\frac{\\pi }{6}[\/latex], Quadrant III, [latex]\\text{sin}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{1}{2}[\/latex], [latex]\\text{cos}\\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>45.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{3\\pi }{4}\\right)=\\frac{\\sqrt{2}}{2}[\/latex], [latex]\\cos \\left(\\frac{4\\pi }{3}\\right)=-\\frac{\\sqrt[]{2}}{2}[\/latex]<\/p>\n<p>47.\u00a0[latex]\\frac{\\pi }{3}[\/latex], Quadrant II, [latex]\\text{sin}\\left(\\frac{2\\pi }{3}\\right)=\\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos \\left(\\frac{2\\pi }{3}\\right)=-\\frac{1}{2}[\/latex]<\/p>\n<p>49.\u00a0[latex]\\frac{\\pi }{4}[\/latex], Quadrant IV, [latex]\\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<p>51.\u00a0[latex]\\frac{\\sqrt{77}}{9}[\/latex]<\/p>\n<p>53.\u00a0[latex]-\\frac{\\sqrt{15}}{4}[\/latex]<\/p>\n<p>55.\u00a0[latex]\\left(-10,10\\sqrt{3}\\right)[\/latex]<\/p>\n<p>57.\u00a0[latex]\\left(-2.778,15.757\\right)[\/latex]<\/p>\n<p>59.\u00a0[latex]\\left[-1,1\\right][\/latex]<\/p>\n<p>61.\u00a0[latex]\\sin t=\\frac{1}{2},\\cos t=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>63.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=-\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p>65.\u00a0[latex]\\sin t=\\frac{\\sqrt{3}}{2},\\cos t=-\\frac{1}{2}[\/latex]<\/p>\n<p>67.\u00a0[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p>69.\u00a0[latex]\\sin t=0,\\cos t=-1[\/latex]<\/p>\n<p>71.\u00a0[latex]\\sin t=-0.596,\\cos t=0.803[\/latex]<\/p>\n<p>73.\u00a0[latex]\\sin t=\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>75.\u00a0[latex]\\sin t=-\\frac{1}{2},\\cos t=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>77.\u00a0[latex]\\sin t=0.761,\\cos t=-0.649[\/latex]<\/p>\n<p>79.\u00a0[latex]\\sin t=1,\\cos t=0[\/latex]<\/p>\n<p>81.\u00a0\u22120.1736<\/p>\n<p>83.\u00a00.9511<\/p>\n<p>85.\u00a0\u22120.7071<\/p>\n<p>87.\u00a0\u22120.1392<\/p>\n<p>89.\u00a0\u22120.7660<\/p>\n<p>91.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<p>93.\u00a0[latex]-\\frac{\\sqrt{6}}{4}[\/latex]<\/p>\n<p>95.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<p>97.\u00a0[latex]\\frac{\\sqrt{2}}{4}[\/latex]<\/p>\n<p>99. 0<\/p>\n<p>101.\u00a0[latex]\\left(0,-1\\right)[\/latex]<\/p>\n<p>103.\u00a037.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13814\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23485,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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