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Mathematics LibreTexts

8.2.1: Polar Coordinates (Exercises)

  • Page ID
    22236
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    section 8.2 exercises

    Convert the given polar coordinates to Cartesian coordinates.

    1. \((7, \dfrac{7\pi}{6})\)

    2. \(6, \dfrac{3pi}{4}\)

    3. \(4, \dfrac{7\pi}{4}\)

    4. \(9, \dfrac{4\pi}{3}\)

    5. \(6, -\dfrac{\pi}{4}\)

    6. \(12, -\dfrac{\pi}{3}\)

    7. \(3, \dfrac{\pi}{2}\)

    8. \(5, \pi\)

    9.\(-3, \dfrac{\pi}{6}\)

    10. \(-2, \dfrac{2\pi}{3}\)

    11. (3, 2)

    12. (7, 1)

    Convert the given Cartesian coordinates to polar coordinates.

    13. (4, 2)

    14. (8, 8)

    15. (-4, 6)

    16. (-5, 1)

    17. (3, -5)

    18. (6, -5)

    19. (-10, -13)

    20. (-4, -7)

    Convert the given Cartesian equation to a polar equation.

    21. \(x = 3\)

    22. \(y = 4\)

    23. \(y = 4x^2\)

    24. \(y = 2x^4\)

    25. \(x^2 = y^2 = 4y\)

    26. \(x^2 = y^2 = 3x\)

    27. \(x^2 - y^2 = x\)

    28. \(x^2 - y^2 = 3y\)

    Convert the given polar equation to a Cartesian equation.

    29. \(r = 3\sin(\theta)\)

    30. \(r = 4\cos(\theta)\)

    31. \(r = \dfrac{4}{\sin(\theta) + 7\cos(\theta)}\)

    32. \(r = \dfrac{6}{\cos(\theta) + 3\sin(\theta)}\)

    33. \(r = 2\sec(\theta)\)

    34. \(r = 3\csc(\theta)\)

    35. \(r = \sqrt{r\cos(\theta) + 2}\)

    36. \(r^2 = 4 \sec(\theta)\csc(\theta)\)

    Match each equation with one of the graphs shown.

    37. \(r = 2 + 2\cos(\theta)\)

    38. \(r = 2 + 2\sin(\theta)\)

    39. \(r = 4 + 3\cos(\theta)\)

    40. \(r = 3 + 4\cos(\theta)\)

    41. \(r = 5\)

    42. \(r = 2\sin(\theta)\)

    屏幕快照 2019-07-21 下午12.07.29.png

    Match each equation with one of the graphs shown.

    43. \(r = \text{log}(\theta)\)

    44. \(r = \theta \cos(\theta)\)

    45. \(r = \cos(\dfrac{\theta}{2})\)

    46. \(r = \sin(\theta)\cos^2(\theta)\)

    47. \(r = 1 + 2\sin(3\theta)\)

    48. \(r = 1 + \sin(2\theta)\)

    屏幕快照 2019-07-21 下午12.09.55.png

    Sketch a graph of the polar equation.

    49. \(r = 3\cos(\theta)\)

    50. \(r = 4\sin(\theta)\)

    51. \(r = 3\sin(2\theta)\)

    52. \(r = 4\sin(4\theta)\)

    53. \(r = 5\sin(3\theta)\)

    54. \(r = 4\sin(5\theta)\)

    55. \(r = 3\cos(2\theta)\)

    56. \(r = 4\cos(4\theta)\)

    57. \(r = 2+ 2\cos(\theta)\)

    58. \(r = 3 + 3\sin(\theta)\)

    59. \(r = 1 + 3\sin(\theta)\)

    60. \(r = 2 + 4 \cos(\theta)\)

    61. \(r = 2\theta\)

    62. \(r = \dfrac{1}{\theta}\)

    63. \(r = 3 + \sec(\theta)\), a conchoid

    64. \(r = \dfrac{1}{\sqrt{\theta}}\), a lituus

    65. \(r = 2\sin(\theta)\tan(\theta)\), a cissoid

    66. \(r = 2\sqrt{1- \sin^2(\theta)}\). a hippopede

    Answer

    1. \((-\dfrac{7\sqrt{3}}{2}, -\dfrac{7}{2})\)

    3. \((2\sqrt{2}, -2\sqrt{2})\)

    5. \((3\sqrt{2}, -3\sqrt{2})\)

    7. (0, 3)

    9. \((-\dfrac{3\sqrt{3}}{2}, -\dfrac{3}{2}\)

    11. (−1.248, 2.728)

    13. \((2\sqrt{5}, 0.464)\)

    15. \((2\sqrt{13}, 2.159)\)

    17. \((\sqrt{34}, 5.253)\)

    19. \((\sqrt{269}, 4.057)\)

    21. \(r = 3\sec(\theta)\)

    23. \(r = \dfrac{\sin(\theta)}{4\cos^(2)(\theta)}\)

    25. \(r = 4\sin(\theta)\)

    27. \(r = \dfrac{\cos(\theta)}{(\cos^(2)(\theta) - \sin^(2)(\theta))}\)

    29. \(x^2 + y^2 = 3y\)

    31. \(y = 7x = 4\)

    33. \(x = 2\)

    35. \(x^2 + y^2 = x + 2\)

    37. A

    39. C

    41. E

    43. C

    45. D

    47. F

    49. Screen Shot 2019-10-14 at 10.22.07 PM.png

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