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11.4E: Exercises

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    Exercises

    In Exercises 1 - 16, plot the point given in polar coordinates and then give three different expressions for the point such that

    1. \(r<0 \text { and } 0 \leq \theta \leq 2 \pi,\)
    2. \(r>0 \text { and } \theta \leq 0\)
    3. \(r>0 \text { and } \theta \geq 2 \pi\)
    1. \(\left(2, \frac{\pi}{3}\right)\)
    2. \(\left(5, \frac{7 \pi}{4}\right)\)
    3. \(\left(\frac{1}{3}, \frac{3 \pi}{2}\right)\)
    4. \(\left(\frac{5}{2}, \frac{5 \pi}{6}\right)\)
    5. \(\left(12,-\frac{7 \pi}{6}\right)\)
    6. \(\left(3,-\frac{5 \pi}{4}\right)\)
    7. \((2 \sqrt{2},-\pi)\)
    8. \(\left(\frac{7}{2},-\frac{13 \pi}{6}\right)\)
    9. \((-20,3 \pi)\)
    10. \(\left(-4, \frac{5 \pi}{4}\right)\)
    11. \(\left(-1, \frac{2 \pi}{3}\right)\)
    12. \(\left(-3, \frac{\pi}{2}\right)\)
    13. \(\left(-3,-\frac{11 \pi}{6}\right)\)
    14. \(\left(-2.5,-\frac{\pi}{4}\right)\)
    15. \(\left(-\sqrt{5},-\frac{4 \pi}{3}\right)\)
    16. \((-\pi,-\pi)\)

    In Exercises 17 - 36, convert the point from polar coordinates into rectangular coordinates.

    1. \(\left(5, \frac{7 \pi}{4}\right)\)
    2. \(\left(2, \frac{\pi}{3}\right)\)
    3. \(\left(11,-\frac{7 \pi}{6}\right)\)
    4. \((-20,3 \pi)\)
    5. \(\left(\frac{3}{5}, \frac{\pi}{2}\right)\)
    6. \(\left(-4, \frac{5 \pi}{6}\right)\)
    7. \(\left(9, \frac{7 \pi}{2}\right)\)
    8. \(\left(-5,-\frac{9 \pi}{4}\right)\)
    9. \(\left(42, \frac{13 \pi}{6}\right)\)
    10. \((-117,117 \pi)\)
    11. \((6, \arctan (2))\)
    12. \((10, \arctan (3))\)
    13. \(\left(-3, \arctan \left(\frac{4}{3}\right)\right)\)
    14. \(\left(5, \arctan \left(-\frac{4}{3}\right)\right)\)
    15. \(\left(2, \pi-\arctan \left(\frac{1}{2}\right)\right)\)
    16. \(\left(-\frac{1}{2}, \pi-\arctan (5)\right)\)
    17. \(\left(-1, \pi+\arctan \left(\frac{3}{4}\right)\right)\)
    18. \(\left(\frac{2}{3}, \pi+\arctan (2 \sqrt{2})\right)\)
    19. \((\pi, \arctan (\pi))\)
    20. \(\left(13, \arctan \left(\frac{12}{5}\right)\right)\)

    In Exercises 37 - 56, convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\).

    1. \((0,5)\)
    2. \((3, \sqrt{3})\)
    3. \((7,-7)\)
    4. \((-3,-\sqrt{3})\)
    5. \((-3,0)\)
    6. \((-\sqrt{2}, \sqrt{2})\)
    7. \((-4,-4 \sqrt{3})\)
    8. \(\left(\frac{\sqrt{3}}{4},-\frac{1}{4}\right)\)
    9. \(\left(-\frac{3}{10},-\frac{3 \sqrt{3}}{10}\right)\)
    10. \((-\sqrt{5},-\sqrt{5})\)
    11. \((6,8)\)
    12. \((\sqrt{5}, 2 \sqrt{5})\)
    13. \((-8,1)\)
    14. \((-2 \sqrt{10}, 6 \sqrt{10})\)
    15. \((-5,-12)\)
    16. \(\left(-\frac{\sqrt{5}}{15},-\frac{2 \sqrt{5}}{15}\right)\)
    17. \((24,-7)\)
    18. \((12,-9)\)
    19. \(\left(\frac{\sqrt{2}}{4}, \frac{\sqrt{6}}{4}\right)\)
    20. \(\left(-\frac{\sqrt{65}}{5}, \frac{2 \sqrt{65}}{5}\right)\)

    In Exercises 57 - 76, convert the equation from rectangular coordinates into polar coordinates. Solve for \(r\) in all but #60 through #63. In Exercises 60 - 63, you need to solve for \(\theta\)

    1. \(x = 6\)
    2. \(x = −3\)
    3. \(y = 7\)
    4. \(y = 0\)
    5. \(y = −x\)
    6. \(y=x \sqrt{3}\)
    7. \(y = 2x\)
    8. \(x^{2}+y^{2}=25\)
    9. \(x^{2}+y^{2}=117\)
    10. \(y = 4x − 19\)
    11. \(x = 3y + 1\)
    12. \(y=-3 x^{2}\)
    13. \(4 x=y^{2}\)
    14. \(x^{2}+y^{2}-2 y=0\)
    15. \(x^{2}-4 x+y^{2}=0\)
    16. \(x^{2}+y^{2}=x\)
    17. \(y^{2}=7 y-x^{2}\)
    18. \((x+2)^{2}+y^{2}=4\)
    19. \(x^{2}+(y-3)^{2}=9\)
    20. \(4 x^{2}+4\left(y-\frac{1}{2}\right)^{2}=1\)

    In Exercises 77 - 96, convert the equation from polar coordinates into rectangular coordinates.

    1. \(r = 7\)
    2. \(r = −3\)
    3. \(r=\sqrt{2}\)
    4. \(\theta=\frac{\pi}{4}\)
    5. \(\theta=\frac{2 \pi}{3}\)
    6. \(\theta=\pi\)
    7. \(\theta=\frac{3 \pi}{2}\)
    8. \(r=4 \cos (\theta)\)
    9. \(5 r=\cos (\theta)\)
    10. \(r=3 \sin (\theta)\)
    11. \(r=-2 \sin (\theta)\)
    12. \(r=7 \sec (\theta)\)
    13. \(12 r=\csc (\theta)\)
    14. \(r=-2 \sec (\theta)\)
    15. \(r=-\sqrt{5} \csc (\theta)\)
    16. \(r=2 \sec (\theta) \tan (\theta)\)
    17. \(r=-\csc (\theta) \cot (\theta)\)
    18. \(r^{2}=\sin (2 \theta)\)
    19. \(r=1-2 \cos (\theta)\)
    20. \(r=1+\sin (\theta)\)
    21. Convert the origin (0, 0) into polar coordinates in four different ways.
    22. With the help of your classmates, use the Law of Cosines to develop a formula for the distance between two points in polar coordinates.

    Answers

    1. \(\begin{aligned}
      &\left(2, \frac{\pi}{3}\right),\left(-2, \frac{4 \pi}{3}\right) \\
      &\left(2,-\frac{5 \pi}{3}\right),\left(2, \frac{7 \pi}{3}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.54.48 PM.png

    2. \(\begin{aligned}
      &\left(5, \frac{7 \pi}{4}\right),\left(-5, \frac{3 \pi}{4}\right) \\
      &\left(5,-\frac{\pi}{4}\right),\left(5, \frac{15 \pi}{4}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.55.30 PM.png

    3. \(\begin{aligned}
      &\left(\frac{1}{3}, \frac{3 \pi}{2}\right),\left(-\frac{1}{3}, \frac{\pi}{2}\right) \\
      &\left(\frac{1}{3},-\frac{\pi}{2}\right),\left(\frac{1}{3}, \frac{7 \pi}{2}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.56.21 PM.png

    4. \(\begin{aligned}
      &\left(\frac{5}{2}, \frac{5 \pi}{6}\right),\left(-\frac{5}{2}, \frac{11 \pi}{6}\right) \\
      &\left(\frac{5}{2},-\frac{7 \pi}{6}\right),\left(\frac{5}{2}, \frac{17 \pi}{6}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.57.16 PM.png

    5. \(\begin{aligned}
      &\left(12,-\frac{7 \pi}{6}\right),\left(-12, \frac{11 \pi}{6}\right) \\
      &\left(12,-\frac{19 \pi}{6}\right),\left(12, \frac{17 \pi}{6}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.58.04 PM.png

    6. \(\begin{aligned}
      &\left(3,-\frac{5 \pi}{4}\right),\left(-3, \frac{7 \pi}{4}\right) \\
      &\left(3,-\frac{13 \pi}{4}\right),\left(3, \frac{11 \pi}{4}\right)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.58.53 PM.png

    7. \(\begin{aligned}
      &(2 \sqrt{2},-\pi),(-2 \sqrt{2}, 0) \\
      &(2 \sqrt{2},-3 \pi),(2 \sqrt{2}, 3 \pi)
      \end{aligned}\)

      Screen Shot 2022-05-30 at 11.59.42 PM.png

    8. \(\begin{aligned}
      &\left(\frac{7}{2},-\frac{13 \pi}{6}\right),\left(-\frac{7}{2}, \frac{5 \pi}{6}\right) \\
      &\left(\frac{7}{2},-\frac{\pi}{6}\right),\left(\frac{7}{2}, \frac{23 \pi}{6}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.00.29 AM.png

    9. \(\begin{aligned}
      &(-20,3 \pi),(-20, \pi) \\
      &(20,-2 \pi),(20,4 \pi)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.01.26 AM.png

    10. \(\begin{aligned}
      &\left(-4, \frac{5 \pi}{4}\right),\left(-4, \frac{13 \pi}{4}\right) \\
      &\left(4,-\frac{7 \pi}{4}\right),\left(4, \frac{9 \pi}{4}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.22.13 AM.png

    11. \(\begin{aligned}
      &\left(-1, \frac{2 \pi}{3}\right),\left(-1, \frac{8 \pi}{3}\right) \\
      &\left(1,-\frac{\pi}{3}\right),\left(1, \frac{11 \pi}{3}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.23.18 AM.png

    12. \(\begin{aligned}
      &\left(-3, \frac{\pi}{2}\right),\left(-3, \frac{5 \pi}{2}\right) \\
      &\left(3,-\frac{\pi}{2}\right),\left(3, \frac{7 \pi}{2}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.24.02 AM.png

    13. \(\begin{aligned}
      &\left(-3,-\frac{11 \pi}{6}\right),\left(-3, \frac{\pi}{6}\right) \\
      &\left(3,-\frac{5 \pi}{6}\right),\left(3, \frac{19 \pi}{6}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.25.07 AM.png

    14. \(\begin{aligned}
      &\left(-2.5,-\frac{\pi}{4}\right),\left(-2.5, \frac{7 \pi}{4}\right) \\
      &\left(2.5,-\frac{5 \pi}{4}\right),\left(2.5, \frac{11 \pi}{4}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.33.57 AM.png

    15. \(\begin{aligned}
      &\left(-\sqrt{5},-\frac{4 \pi}{3}\right),\left(-\sqrt{5}, \frac{2 \pi}{3}\right) \\
      &\left(\sqrt{5},-\frac{\pi}{3}\right),\left(\sqrt{5}, \frac{11 \pi}{3}\right)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.34.41 AM.png

    16. \(\begin{aligned}
      &(-\pi,-\pi),(-\pi, \pi) \\
      &(\pi,-2 \pi),(\pi, 2 \pi)
      \end{aligned}\)

      Screen Shot 2022-05-31 at 12.35.31 AM.png

    17. \(\left(\frac{5 \sqrt{2}}{2},-\frac{5 \sqrt{2}}{2}\right)\)
    18. \((1, \sqrt{3})\)
    19. \(\left(-\frac{11 \sqrt{3}}{2}, \frac{11}{2}\right)\)
    20. \((20, 0)\)
    21. \(\left(0, \frac{3}{5}\right)\)
    22. \((2 \sqrt{3},-2)\)
    23. \((0, −9)\)
    24. \(\left(-\frac{5 \sqrt{2}}{2}, \frac{5 \sqrt{2}}{2}\right)\)
    25. \((21 \sqrt{3}, 21)\)
    26. \((117,0)\)
    27. \(\left(\frac{6 \sqrt{5}}{5}, \frac{12 \sqrt{5}}{5}\right)\)
    28. \((\sqrt{10}, 3 \sqrt{10})\)
    29. \(\left(-\frac{9}{5},-\frac{12}{5}\right)\)
    30. \((3,-4)\)
    31. \(\left(-\frac{4 \sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right)\)
    32. \(\left(\frac{\sqrt{26}}{52},-\frac{5 \sqrt{26}}{52}\right)\)
    33. \(\left(\frac{4}{5}, \frac{3}{5}\right)\)
    34. \(\left(-\frac{2}{9},-\frac{4 \sqrt{2}}{9}\right)\)
    35. \(\left(\frac{\pi}{\sqrt{1+\pi^{2}}}, \frac{\pi^{2}}{\sqrt{1+\pi^{2}}}\right)\)
    36. \((5,12)\)
    37. \(\left(5, \frac{\pi}{2}\right)\)
    38. \(\left(2 \sqrt{3}, \frac{\pi}{6}\right)\)
    39. \(\left(7 \sqrt{2}, \frac{7 \pi}{4}\right)\)
    40. \(\left(2 \sqrt{3}, \frac{7 \pi}{6}\right)\)
    41. \((3, \pi)\)
    42. \(\left(2, \frac{3 \pi}{4}\right)\)
    43. \(\left(8, \frac{4 \pi}{3}\right)\)
    44. \(\left(\frac{1}{2}, \frac{11 \pi}{6}\right)\)
    45. \(\left(\frac{3}{5}, \frac{4 \pi}{3}\right)\)
    46. \(\left(\sqrt{10}, \frac{5 \pi}{4}\right)\)
    47. \(\left(10, \arctan \left(\frac{4}{3}\right)\right)\)
    48. \((5, \arctan (2))\)
    49. \(\left(\sqrt{65}, \pi-\arctan \left(\frac{1}{8}\right)\right)\)
    50. \((20, \pi-\arctan (3))\)
    51. \(\left(13, \pi+\arctan \left(\frac{12}{5}\right)\right)\)
    52. \(\left(\frac{1}{3}, \pi+\arctan (2)\right)\)
    53. \(\left(25,2 \pi-\arctan \left(\frac{7}{24}\right)\right)\)
    54. \(\left(15,2 \pi-\arctan \left(\frac{3}{4}\right)\right)\)
    55. \(\left(\frac{\sqrt{2}}{2}, \frac{\pi}{3}\right)\)
    56. \((\sqrt{13}, \pi-\arctan (2))\)
    57. \(r=6 \sec (\theta)\)
    58. \(r=-3 \sec (\theta)\)
    59. \(r=7 \csc (\theta)\)
    60. \(\theta=0\)
    61. \(\theta=\frac{3 \pi}{4}\)
    62. \(\theta=\frac{\pi}{3}\)
    63. \(\theta=\arctan (2)\)
    64. \(r=5\)
    65. \(r=\sqrt{117}\)
    66. \(r=\frac{19}{4 \cos (\theta)-\sin (\theta)}\)
    67. \(x=\frac{1}{\cos (\theta)-3 \sin (\theta)}\)
    68. \(r=\frac{-\sec (\theta) \tan (\theta)}{3}\)
    69. \(r=4 \csc (\theta) \cot (\theta)\)
    70. \(r=2 \sin (\theta)\)
    71. \(r=4 \cos (\theta)\)
    72. \(r=\cos (\theta)\)
    73. \(r=7 \sin (\theta)\)
    74. \(r=-4 \cos (\theta)\)
    75. \(r=6 \sin (\theta)\)
    76. \(r=\sin (\theta)\)
    77. \(x^{2}+y^{2}=49\)
    78. \(x^{2}+y^{2}=9\)
    79. \(x^{2}+y^{2}=2\)
    80. \(y = x\)
    81. \(y=-\sqrt{3} x\)
    82. \(y = 0\)
    83. \(x = 0\)
    84. \(x^{2}+y^{2}=4 x \text { or }(x-2)^{2}+y^{2}=4\)
    85. \(5 x^{2}+5 y^{2}=x \text { or }\left(x-\frac{1}{10}\right)^{2}+y^{2}=\frac{1}{100}\)
    86. \(x^{2}+y^{2}=3 y \text { or } x^{2}+\left(y-\frac{3}{2}\right)^{2}=\frac{9}{4}\)
    87. \(x^{2}+y^{2}=-2 y \text { or } x^{2}+(y+1)^{2}=1\)
    88. \(x = 7\)
    89. \(y=\frac{1}{12}\)
    90. \(x = −2\)
    91. \(y=-\sqrt{5}\)
    92. \(x^{2}=2 y\)
    93. \(y^{2}=-x\)
    94. \(\left(x^{2}+y^{2}\right)^{2}=2 x y\)
    95. \(\left(x^{2}+2 x+y^{2}\right)^{2}=x^{2}+y^{2}\)
    96. \(\left(x^{2}+y^{2}+y\right)^{2}=x^{2}+y^{2}\)
    97. Any point of the form \((0, \theta)\) will work, e.g. \((0, \pi),(0,-117),\left(0, \frac{23 \pi}{4}\right) \text { and }(0,0)\).

    11.4E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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