Problem Set 59: Polar Coordinates

1. How are polar coordinates different from rectangular coordinates?

2. How are the polar axes different from the x– and y-axes of the Cartesian plane?

3. Explain how polar coordinates are graphed.

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

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.

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.

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

7. [latex]\left(5,\pi \right)[/latex]

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

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

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

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. 11. [latex]\left(4,2\right)[/latex] 12. [latex]\left(-4,6\right)[/latex] 13. [latex]\left(3,-5\right)[/latex] 14. [latex]\left(-10,-13\right)[/latex] 15. [latex]\left(8,8\right)[/latex] For the following exercises, convert the given Cartesian equation to a polar equation. 16. [latex]x=3[/latex] 17. [latex]y=4[/latex] 18. [latex]y=4{x}^{2}[/latex] 19. [latex]y=2{x}^{4}[/latex] 20. [latex]{x}^{2}+{y}^{2}=4y[/latex] 21. [latex]{x}^{2}+{y}^{2}=3x[/latex] 22. [latex]{x}^{2}-{y}^{2}=x[/latex] 23. [latex]{x}^{2}-{y}^{2}=3y[/latex] 24. [latex]{x}^{2}+{y}^{2}=9[/latex] 25. [latex]{x}^{2}=9y[/latex] 26. [latex]{y}^{2}=9x[/latex] 27. [latex]9xy=1[/latex] For 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. 28. [latex]r=3\sin \theta[/latex] 29. [latex]r=4\cos \theta[/latex] 30. [latex]r=\frac{4}{\sin \theta +7\cos \theta }[/latex] 31. [latex]r=\frac{6}{\cos \theta +3\sin \theta }[/latex] 32. [latex]r=2\sec \theta[/latex] 33. [latex]r=3\csc \theta[/latex] 34. [latex]r=\sqrt{r\cos \theta +2}[/latex] 35. [latex]{r}^{2}=4\sec \theta \csc \theta[/latex] 36. [latex]r=4[/latex] 37. [latex]{r}^{2}=4[/latex] 38. [latex]r=\frac{1}{4\cos \theta -3\sin \theta }[/latex] 39. [latex]r=\frac{3}{\cos \theta -5\sin \theta }[/latex] For the following exercises, find the polar coordinates of the point. 40. Polar coordinate system with a point located on the third concentric circle and pi/2.

41.
Polar coordinate system with a point located on the third concentric circle and midway between pi/2 and pi in the second quadrant.

42.
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).

43.
Polar coordinate system with a point located on the fifth concentric circle and pi.

44.
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).

For the following exercises, plot the points.

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

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

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

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

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

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

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

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

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

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

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

55. [latex]5x-y=6[/latex]

56. [latex]2x+7y=-3[/latex]

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

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

59. [latex]x=2[/latex]

60. [latex]{x}^{2}+{y}^{2}=5y[/latex]

61. [latex]{x}^{2}+{y}^{2}=3x[/latex]

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

62. [latex]r=6[/latex]

63. [latex]r=-4[/latex]

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

65. [latex]\theta =\frac{\pi }{4}[/latex]

66. [latex]r=\sec \theta[/latex]

67. [latex]r=-10\sin \theta[/latex]

68. [latex]r=3\cos \theta[/latex]

69. Use a graphing calculator to find the rectangular coordinates of [latex]\left(2,-\frac{\pi }{5}\right)[/latex]. Round to the nearest thousandth.

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.

71. Use a graphing calculator to find the polar coordinates of [latex]\left(-7,8\right)[/latex] in degrees. Round to the nearest thousandth.

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

73. Use a graphing calculator to find the polar coordinates of [latex]\left(-2,0\right)[/latex] in radians. Round to the nearest hundredth.

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

75. Describe the graph of [latex]r=a\sec \theta ;a<0[/latex]. 76. Describe the graph of [latex]r=a\csc \theta ;a>0[/latex].

77. Describe the graph of [latex]r=a\csc \theta ;a<0[/latex]. 78. What polar equations will give an oblique line? For the following exercises, graph the polar inequality. 79. [latex]r<4[/latex] 80. [latex]0\le \theta \le \frac{\pi }{4}[/latex] 81. [latex]\theta =\frac{\pi }{4},r\ge 2[/latex] 82. [latex]\theta =\frac{\pi }{4},r\ge -3[/latex] 83. [latex]0\le \theta \le \frac{\pi }{3},r<2[/latex] 84. [latex]\frac{-\pi }{6}<\theta \le \frac{\pi }{3},-3