{"id":15725,"date":"2019-09-05T18:39:07","date_gmt":"2019-09-05T18:39:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=15725"},"modified":"2025-02-05T05:24:19","modified_gmt":"2025-02-05T05:24:19","slug":"problem-set-59-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-59-polar-coordinates\/","title":{"raw":"Problem Set 59: Polar Coordinates","rendered":"Problem Set 59: Polar Coordinates"},"content":{"raw":"<dl id=\"fs-id1165134149787\" class=\"definition\">\r\n \t<dd>1. How are polar coordinates different from rectangular coordinates?\r\n\r\n2.\u00a0How are the polar axes different from the <em>x<\/em>- and <em>y<\/em>-axes of the Cartesian plane?\r\n\r\n3. Explain how polar coordinates are graphed.\r\n\r\n4.\u00a0How are the points [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] related?\r\n\r\n5. Explain why the points [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] are the same.\r\n\r\nFor the following exercises, convert the given polar coordinates to Cartesian coordinates with [latex]r&gt;0[\/latex] and [latex]0\\le \\theta \\le 2\\pi [\/latex]. Remember to consider the quadrant in which the given point is located when determining [latex]\\theta [\/latex] for the point.\r\n\r\n6. [latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]\r\n\r\n7. [latex]\\left(5,\\pi \\right)[\/latex]\r\n\r\n8.\u00a0[latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]\r\n\r\n9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]\r\n\r\n10.\u00a0[latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]\r\n\r\nFor the following exercises, convert the given Cartesian coordinates to polar coordinates with [latex]r&gt;0,0\\le \\theta &lt;2\\pi [\/latex]. Remember to consider the quadrant in which the given point is located.\r\n\r\n11. [latex]\\left(4,2\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\left(-4,6\\right)[\/latex]\r\n\r\n13. [latex]\\left(3,-5\\right)[\/latex]\r\n\r\n14.\u00a0[latex]\\left(-10,-13\\right)[\/latex]\r\n\r\n15. [latex]\\left(8,8\\right)[\/latex]\r\n\r\nFor the following exercises, convert the given Cartesian equation to a polar equation.\r\n\r\n16. [latex]x=3[\/latex]\r\n\r\n17. [latex]y=4[\/latex]\r\n\r\n18.\u00a0[latex]y=4{x}^{2}[\/latex]\r\n\r\n19. [latex]y=2{x}^{4}[\/latex]\r\n\r\n20.\u00a0[latex]{x}^{2}+{y}^{2}=4y[\/latex]\r\n\r\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\r\n\r\n22.\u00a0[latex]{x}^{2}-{y}^{2}=x[\/latex]\r\n\r\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\r\n\r\n24.\u00a0[latex]{x}^{2}+{y}^{2}=9[\/latex]\r\n\r\n25. [latex]{x}^{2}=9y[\/latex]\r\n\r\n26.\u00a0[latex]{y}^{2}=9x[\/latex]\r\n\r\n27. [latex]9xy=1[\/latex]\r\n\r\nFor the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.\r\n\r\n28. [latex]r=3\\sin \\theta [\/latex]\r\n\r\n29. [latex]r=4\\cos \\theta [\/latex]\r\n\r\n30.\u00a0[latex]r=\\frac{4}{\\sin \\theta +7\\cos \\theta }[\/latex]\r\n\r\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\r\n\r\n32.\u00a0[latex]r=2\\sec \\theta [\/latex]\r\n\r\n33. [latex]r=3\\csc \\theta [\/latex]\r\n\r\n34.\u00a0[latex]r=\\sqrt{r\\cos \\theta +2}[\/latex]\r\n\r\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta [\/latex]\r\n\r\n36.\u00a0[latex]r=4[\/latex]\r\n\r\n37. [latex]{r}^{2}=4[\/latex]\r\n\r\n38.\u00a0[latex]r=\\frac{1}{4\\cos \\theta -3\\sin \\theta }[\/latex]\r\n\r\n39. [latex]r=\\frac{3}{\\cos \\theta -5\\sin \\theta }[\/latex]\r\n\r\nFor the following exercises, find the polar coordinates of the point.\r\n\r\n40.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165415\/CNX_Precalc_Figure_08_03_201n2.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and pi\/2.\" \/>\r\n\r\n41.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165417\/CNX_Precalc_Figure_08_03_202n2.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and midway between pi\/2 and pi in the second quadrant. \" \/>\r\n\r\n42.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165419\/CNX_Precalc_Figure_08_03_203n2.jpg\" alt=\"Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi\/2 (closer to pi).\" \/>\r\n\r\n43.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165421\/CNX_Precalc_Figure_08_03_204n2.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi.\" \/>\r\n\r\n44.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165423\/CNX_Precalc_Figure_08_03_205n2.jpg\" alt=\"Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi\/2 and 2pi (closer to 3pi\/2).\" \/>\r\n\r\nFor the following exercises, plot the points.\r\n\r\n45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]\r\n\r\n46. [latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]\r\n\r\n47. [latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]\r\n\r\n48. [latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]\r\n\r\n49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]\r\n\r\n50. [latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]\r\n\r\n51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]\r\n\r\n52. [latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]\r\n\r\n53. [latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]\r\n\r\n54. [latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]\r\n\r\nFor the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.\r\n\r\n55. [latex]5x-y=6[\/latex]\r\n\r\n56. [latex]2x+7y=-3[\/latex]\r\n\r\n57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]\r\n\r\n58. [latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]\r\n\r\n59. [latex]x=2[\/latex]\r\n\r\n60. [latex]{x}^{2}+{y}^{2}=5y[\/latex]\r\n\r\n61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\r\n\r\nFor the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.\r\n\r\n62. [latex]r=6[\/latex]\r\n\r\n63. [latex]r=-4[\/latex]\r\n\r\n64. [latex]\\theta =-\\frac{2\\pi }{3}[\/latex]\r\n\r\n65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]\r\n\r\n66. [latex]r=\\sec \\theta [\/latex]\r\n\r\n67. [latex]r=-10\\sin \\theta [\/latex]\r\n\r\n68. [latex]r=3\\cos \\theta [\/latex]\r\n\r\n69. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(2,-\\frac{\\pi }{5}\\right)[\/latex]. Round to the nearest thousandth.\r\n\r\n70.\u00a0Use a graphing calculator to find the rectangular coordinates of [latex]\\left(-3,\\frac{3\\pi }{7}\\right)[\/latex]. Round to the nearest thousandth.\r\n\r\n71. Use a graphing calculator to find the polar coordinates of [latex]\\left(-7,8\\right)[\/latex] in degrees. Round to the nearest thousandth.\r\n\r\n72.\u00a0Use a graphing calculator to find the polar coordinates of [latex]\\left(3,-4\\right)[\/latex] in degrees. Round to the nearest hundredth.\r\n\r\n73. Use a graphing calculator to find the polar coordinates of [latex]\\left(-2,0\\right)[\/latex] in radians. Round to the nearest hundredth.\r\n\r\n74.\u00a0Describe the graph of [latex]r=a\\sec \\theta ;a&gt;0[\/latex].\r\n\r\n75. Describe the graph of [latex]r=a\\sec \\theta ;a&lt;0[\/latex].\r\n\r\n76.\u00a0Describe the graph of [latex]r=a\\csc \\theta ;a&gt;0[\/latex].\r\n\r\n77. Describe the graph of [latex]r=a\\csc \\theta ;a&lt;0[\/latex].\r\n\r\n78.\u00a0What polar equations will give an oblique line?\r\n\r\nFor the following exercises, graph the polar inequality.\r\n\r\n79. [latex]r&lt;4[\/latex]\r\n\r\n80. [latex]0\\le \\theta \\le \\frac{\\pi }{4}[\/latex]\r\n\r\n81. [latex]\\theta =\\frac{\\pi }{4},r\\ge 2[\/latex]\r\n\r\n82. [latex]\\theta =\\frac{\\pi }{4},r\\ge -3[\/latex]\r\n\r\n83. [latex]0\\le \\theta \\le \\frac{\\pi }{3},r&lt;2[\/latex]\r\n\r\n84. [latex]\\frac{-\\pi }{6}&lt;\\theta \\le \\frac{\\pi }{3},-3&lt;r&lt;2[\/latex]<\/dd>\r\n<\/dl>","rendered":"<dl id=\"fs-id1165134149787\" class=\"definition\">\n<dd>1. How are polar coordinates different from rectangular coordinates?<\/p>\n<p>2.\u00a0How are the polar axes different from the <em>x<\/em>&#8211; and <em>y<\/em>-axes of the Cartesian plane?<\/p>\n<p>3. Explain how polar coordinates are graphed.<\/p>\n<p>4.\u00a0How are the points [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] related?<\/p>\n<p>5. Explain why the points [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(3,-\\frac{\\pi }{2}\\right)[\/latex] are the same.<\/p>\n<p>For the following exercises, convert the given polar coordinates to Cartesian coordinates with [latex]r>0[\/latex] and [latex]0\\le \\theta \\le 2\\pi[\/latex]. Remember to consider the quadrant in which the given point is located when determining [latex]\\theta[\/latex] for the point.<\/p>\n<p>6. [latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<p>7. [latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n<p>8.\u00a0[latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<p>9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<p>10.\u00a0[latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n<p>For the following exercises, convert the given Cartesian coordinates to polar coordinates with [latex]r>0,0\\le \\theta <2\\pi[\/latex]. Remember to consider the quadrant in which the given point is located.\n\n11. [latex]\\left(4,2\\right)[\/latex]\n\n12.\u00a0[latex]\\left(-4,6\\right)[\/latex]\n\n13. [latex]\\left(3,-5\\right)[\/latex]\n\n14.\u00a0[latex]\\left(-10,-13\\right)[\/latex]\n\n15. [latex]\\left(8,8\\right)[\/latex]\n\nFor the following exercises, convert the given Cartesian equation to a polar equation.\n\n16. [latex]x=3[\/latex]\n\n17. [latex]y=4[\/latex]\n\n18.\u00a0[latex]y=4{x}^{2}[\/latex]\n\n19. [latex]y=2{x}^{4}[\/latex]\n\n20.\u00a0[latex]{x}^{2}+{y}^{2}=4y[\/latex]\n\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\n\n22.\u00a0[latex]{x}^{2}-{y}^{2}=x[\/latex]\n\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\n\n24.\u00a0[latex]{x}^{2}+{y}^{2}=9[\/latex]\n\n25. [latex]{x}^{2}=9y[\/latex]\n\n26.\u00a0[latex]{y}^{2}=9x[\/latex]\n\n27. [latex]9xy=1[\/latex]\n\nFor the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.\n\n28. [latex]r=3\\sin \\theta[\/latex]\n\n29. [latex]r=4\\cos \\theta[\/latex]\n\n30.\u00a0[latex]r=\\frac{4}{\\sin \\theta +7\\cos \\theta }[\/latex]\n\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\n\n32.\u00a0[latex]r=2\\sec \\theta[\/latex]\n\n33. [latex]r=3\\csc \\theta[\/latex]\n\n34.\u00a0[latex]r=\\sqrt{r\\cos \\theta +2}[\/latex]\n\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta[\/latex]\n\n36.\u00a0[latex]r=4[\/latex]\n\n37. [latex]{r}^{2}=4[\/latex]\n\n38.\u00a0[latex]r=\\frac{1}{4\\cos \\theta -3\\sin \\theta }[\/latex]\n\n39. [latex]r=\\frac{3}{\\cos \\theta -5\\sin \\theta }[\/latex]\n\nFor the following exercises, find the polar coordinates of the point.\n\n40.\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165415\/CNX_Precalc_Figure_08_03_201n2.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and pi\/2.\" \/><\/p>\n<p>41.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165417\/CNX_Precalc_Figure_08_03_202n2.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and midway between pi\/2 and pi in the second quadrant.\" \/><\/p>\n<p>42.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165419\/CNX_Precalc_Figure_08_03_203n2.jpg\" alt=\"Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi\/2 (closer to pi).\" \/><\/p>\n<p>43.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165421\/CNX_Precalc_Figure_08_03_204n2.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi.\" \/><\/p>\n<p>44.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27165423\/CNX_Precalc_Figure_08_03_205n2.jpg\" alt=\"Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi\/2 and 2pi (closer to 3pi\/2).\" \/><\/p>\n<p>For the following exercises, plot the points.<\/p>\n<p>45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<p>46. [latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<p>47. [latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n<p>48. [latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<p>49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<p>50. [latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]<\/p>\n<p>51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n<p>52. [latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<p>53. [latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<p>54. [latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n<p>For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.<\/p>\n<p>55. [latex]5x-y=6[\/latex]<\/p>\n<p>56. [latex]2x+7y=-3[\/latex]<\/p>\n<p>57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]<\/p>\n<p>58. [latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]<\/p>\n<p>59. [latex]x=2[\/latex]<\/p>\n<p>60. [latex]{x}^{2}+{y}^{2}=5y[\/latex]<\/p>\n<p>61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/p>\n<p>For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.<\/p>\n<p>62. [latex]r=6[\/latex]<\/p>\n<p>63. [latex]r=-4[\/latex]<\/p>\n<p>64. [latex]\\theta =-\\frac{2\\pi }{3}[\/latex]<\/p>\n<p>65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/p>\n<p>66. [latex]r=\\sec \\theta[\/latex]<\/p>\n<p>67. [latex]r=-10\\sin \\theta[\/latex]<\/p>\n<p>68. [latex]r=3\\cos \\theta[\/latex]<\/p>\n<p>69. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(2,-\\frac{\\pi }{5}\\right)[\/latex]. Round to the nearest thousandth.<\/p>\n<p>70.\u00a0Use a graphing calculator to find the rectangular coordinates of [latex]\\left(-3,\\frac{3\\pi }{7}\\right)[\/latex]. Round to the nearest thousandth.<\/p>\n<p>71. Use a graphing calculator to find the polar coordinates of [latex]\\left(-7,8\\right)[\/latex] in degrees. Round to the nearest thousandth.<\/p>\n<p>72.\u00a0Use a graphing calculator to find the polar coordinates of [latex]\\left(3,-4\\right)[\/latex] in degrees. Round to the nearest hundredth.<\/p>\n<p>73. Use a graphing calculator to find the polar coordinates of [latex]\\left(-2,0\\right)[\/latex] in radians. Round to the nearest hundredth.<\/p>\n<p>74.\u00a0Describe the graph of [latex]r=a\\sec \\theta ;a>0[\/latex].<\/p>\n<p>75. Describe the graph of [latex]r=a\\sec \\theta ;a<0[\/latex].\n\n76.\u00a0Describe the graph of [latex]r=a\\csc \\theta ;a>0[\/latex].<\/p>\n<p>77. Describe the graph of [latex]r=a\\csc \\theta ;a<0[\/latex].\n\n78.\u00a0What polar equations will give an oblique line?\n\nFor the following exercises, graph the polar inequality.\n\n79. [latex]r<4[\/latex]\n\n80. [latex]0\\le \\theta \\le \\frac{\\pi }{4}[\/latex]\n\n81. [latex]\\theta =\\frac{\\pi }{4},r\\ge 2[\/latex]\n\n82. [latex]\\theta =\\frac{\\pi }{4},r\\ge -3[\/latex]\n\n83. [latex]0\\le \\theta \\le \\frac{\\pi }{3},r<2[\/latex]\n\n84. [latex]\\frac{-\\pi }{6}<\\theta \\le \\frac{\\pi }{3},-3<r<2[\/latex]<\/dd>\n<\/dl>\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-15725\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <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>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":169554,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15725","chapter","type-chapter","status-publish","hentry"],"part":14256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15725","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15725\/revisions"}],"predecessor-version":[{"id":15727,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15725\/revisions\/15727"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14256"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15725\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=15725"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=15725"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=15725"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=15725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}