{"id":1417,"date":"2023-06-05T14:51:32","date_gmt":"2023-06-05T14:51:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-59-polar-coordinates\/"},"modified":"2023-06-05T14:51:32","modified_gmt":"2023-06-05T14:51:32","slug":"problem-set-59-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-59-polar-coordinates\/","title":{"raw":"Problem Set 59: Polar Coordinates","rendered":"Problem Set 59: Polar Coordinates"},"content":{"raw":"\n
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1. How are polar coordinates different from rectangular coordinates?\n\n2. How are the polar axes different from the x<\/em>- and y<\/em>-axes of the Cartesian plane?\n\n3. Explain how polar coordinates are graphed.\n\n4. How are the points [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] related?\n\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.\n\nFor 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.\n\n6. [latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]\n\n7. [latex]\\left(5,\\pi \\right)[\/latex]\n\n8. [latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]\n\n9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]\n\n10. [latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]\n\nFor 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. [latex]\\left(-4,6\\right)[\/latex]\n\n13. [latex]\\left(3,-5\\right)[\/latex]\n\n14. [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. [latex]y=4{x}^{2}[\/latex]\n\n19. [latex]y=2{x}^{4}[\/latex]\n\n20. [latex]{x}^{2}+{y}^{2}=4y[\/latex]\n\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\n\n22. [latex]{x}^{2}-{y}^{2}=x[\/latex]\n\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\n\n24. [latex]{x}^{2}+{y}^{2}=9[\/latex]\n\n25. [latex]{x}^{2}=9y[\/latex]\n\n26. [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. [latex]r=\\frac{4}{\\sin \\theta +7\\cos \\theta }[\/latex]\n\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\n\n32. [latex]r=2\\sec \\theta [\/latex]\n\n33. [latex]r=3\\csc \\theta [\/latex]\n\n34. [latex]r=\\sqrt{r\\cos \\theta +2}[\/latex]\n\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta [\/latex]\n\n36. [latex]r=4[\/latex]\n\n37. [latex]{r}^{2}=4[\/latex]\n\n38. [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\"Polar\n\n41.\n\"Polar\n\n42.\n\"Polar\n\n43.\n\"Polar\n\n44.\n\"Polar\n\nFor the following exercises, plot the points.\n\n45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]\n\n46. [latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]\n\n47. [latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]\n\n48. [latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]\n\n49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]\n\n50. [latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]\n\n51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]\n\n52. [latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]\n\n53. [latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]\n\n54. [latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]\n\nFor the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.\n\n55. [latex]5x-y=6[\/latex]\n\n56. [latex]2x+7y=-3[\/latex]\n\n57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]\n\n58. [latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]\n\n59. [latex]x=2[\/latex]\n\n60. [latex]{x}^{2}+{y}^{2}=5y[\/latex]\n\n61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\n\nFor the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.\n\n62. [latex]r=6[\/latex]\n\n63. [latex]r=-4[\/latex]\n\n64. [latex]\\theta =-\\frac{2\\pi }{3}[\/latex]\n\n65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]\n\n66. [latex]r=\\sec \\theta [\/latex]\n\n67. [latex]r=-10\\sin \\theta [\/latex]\n\n68. [latex]r=3\\cos \\theta [\/latex]\n\n69. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(2,-\\frac{\\pi }{5}\\right)[\/latex]. Round to the nearest thousandth.\n\n70. Use a graphing calculator to find the rectangular coordinates of [latex]\\left(-3,\\frac{3\\pi }{7}\\right)[\/latex]. Round to the nearest thousandth.\n\n71. Use a graphing calculator to find the polar coordinates of [latex]\\left(-7,8\\right)[\/latex] in degrees. Round to the nearest thousandth.\n\n72. Use a graphing calculator to find the polar coordinates of [latex]\\left(3,-4\\right)[\/latex] in degrees. Round to the nearest hundredth.\n\n73. Use a graphing calculator to find the polar coordinates of [latex]\\left(-2,0\\right)[\/latex] in radians. Round to the nearest hundredth.\n\n74. Describe the graph of [latex]r=a\\sec \\theta ;a>0[\/latex].\n\n75. Describe the graph of [latex]r=a\\sec \\theta ;a<0[\/latex].\n\n76. Describe the graph of [latex]r=a\\csc \\theta ;a>0[\/latex].\n\n77. Describe the graph of [latex]r=a\\csc \\theta ;a<0[\/latex].\n\n78. What 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","rendered":"
\n
1. How are polar coordinates different from rectangular coordinates?<\/p>\n

2. How are the polar axes different from the x<\/em>– and y<\/em>-axes of the Cartesian plane?<\/p>\n

3. Explain how polar coordinates are graphed.<\/p>\n

4. How are the points [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] and [latex]\\left(-3,\\frac{\\pi }{2}\\right)[\/latex] related?<\/p>\n

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

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

6. [latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n

7. [latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n

8. [latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n

9. [latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n

10. [latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n

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. [latex]\\left(-4,6\\right)[\/latex]\n\n13. [latex]\\left(3,-5\\right)[\/latex]\n\n14. [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. [latex]y=4{x}^{2}[\/latex]\n\n19. [latex]y=2{x}^{4}[\/latex]\n\n20. [latex]{x}^{2}+{y}^{2}=4y[\/latex]\n\n21. [latex]{x}^{2}+{y}^{2}=3x[\/latex]\n\n22. [latex]{x}^{2}-{y}^{2}=x[\/latex]\n\n23. [latex]{x}^{2}-{y}^{2}=3y[\/latex]\n\n24. [latex]{x}^{2}+{y}^{2}=9[\/latex]\n\n25. [latex]{x}^{2}=9y[\/latex]\n\n26. [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. [latex]r=\\frac{4}{\\sin \\theta +7\\cos \\theta }[\/latex]\n\n31. [latex]r=\\frac{6}{\\cos \\theta +3\\sin \\theta }[\/latex]\n\n32. [latex]r=2\\sec \\theta[\/latex]\n\n33. [latex]r=3\\csc \\theta[\/latex]\n\n34. [latex]r=\\sqrt{r\\cos \\theta +2}[\/latex]\n\n35. [latex]{r}^{2}=4\\sec \\theta \\csc \\theta[\/latex]\n\n36. [latex]r=4[\/latex]\n\n37. [latex]{r}^{2}=4[\/latex]\n\n38. [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\"Polar<\/p>\n

41.
\n\"Polar<\/p>\n

42.
\n\"Polar<\/p>\n

43.
\n\"Polar<\/p>\n

44.
\n\"Polar<\/p>\n

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

45. [latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n

46. [latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n

47. [latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n

48. [latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n

49. [latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n

50. [latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]<\/p>\n

51. [latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n

52. [latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n

53. [latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n

54. [latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.<\/p>\n

55. [latex]5x-y=6[\/latex]<\/p>\n

56. [latex]2x+7y=-3[\/latex]<\/p>\n

57. [latex]{x}^{2}+{\\left(y - 1\\right)}^{2}=1[\/latex]<\/p>\n

58. [latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]<\/p>\n

59. [latex]x=2[\/latex]<\/p>\n

60. [latex]{x}^{2}+{y}^{2}=5y[\/latex]<\/p>\n

61. [latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/p>\n

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.<\/p>\n

62. [latex]r=6[\/latex]<\/p>\n

63. [latex]r=-4[\/latex]<\/p>\n

64. [latex]\\theta =-\\frac{2\\pi }{3}[\/latex]<\/p>\n

65. [latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/p>\n

66. [latex]r=\\sec \\theta[\/latex]<\/p>\n

67. [latex]r=-10\\sin \\theta[\/latex]<\/p>\n

68. [latex]r=3\\cos \\theta[\/latex]<\/p>\n

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

70. Use 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

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

72. Use a graphing calculator to find the polar coordinates of [latex]\\left(3,-4\\right)[\/latex] in degrees. Round to the nearest hundredth.<\/p>\n

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

74. Describe the graph of [latex]r=a\\sec \\theta ;a>0[\/latex].<\/p>\n

75. Describe the graph of [latex]r=a\\sec \\theta ;a<0[\/latex].\n\n76. Describe the graph of [latex]r=a\\csc \\theta ;a>0[\/latex].<\/p>\n

77. Describe the graph of [latex]r=a\\csc \\theta ;a<0[\/latex].\n\n78. What 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\n<\/dl>\n\n\t\t\t

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