## 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 $\left(3,\frac{\pi }{2}\right)$ and $\left(-3,\frac{\pi }{2}\right)$ related?

5. Explain why the points $\left(-3,\frac{\pi }{2}\right)$ and $\left(3,-\frac{\pi }{2}\right)$ are the same.

For the following exercises, convert the given polar coordinates to Cartesian coordinates with $r>0$ and $0\le \theta \le 2\pi$. Remember to consider the quadrant in which the given point is located when determining $\theta$ for the point.

6. $\left(7,\frac{7\pi }{6}\right)$

7. $\left(5,\pi \right)$

8. $\left(6,-\frac{\pi }{4}\right)$

9. $\left(-3,\frac{\pi }{6}\right)$

10. $\left(4,\frac{7\pi }{4}\right)$

For the following exercises, convert the given Cartesian coordinates to polar coordinates with $r>0,0\le \theta <2\pi$. Remember to consider the quadrant in which the given point is located.

11. $\left(4,2\right)$

12. $\left(-4,6\right)$

13. $\left(3,-5\right)$

14. $\left(-10,-13\right)$

15. $\left(8,8\right)$

For the following exercises, convert the given Cartesian equation to a polar equation.

16. $x=3$

17. $y=4$

18. $y=4{x}^{2}$

19. $y=2{x}^{4}$

20. ${x}^{2}+{y}^{2}=4y$

21. ${x}^{2}+{y}^{2}=3x$

22. ${x}^{2}-{y}^{2}=x$

23. ${x}^{2}-{y}^{2}=3y$

24. ${x}^{2}+{y}^{2}=9$

25. ${x}^{2}=9y$

26. ${y}^{2}=9x$

27. $9xy=1$

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. $r=3\sin \theta$

29. $r=4\cos \theta$

30. $r=\frac{4}{\sin \theta +7\cos \theta }$

31. $r=\frac{6}{\cos \theta +3\sin \theta }$

32. $r=2\sec \theta$

33. $r=3\csc \theta$

34. $r=\sqrt{r\cos \theta +2}$

35. ${r}^{2}=4\sec \theta \csc \theta$

36. $r=4$

37. ${r}^{2}=4$

38. $r=\frac{1}{4\cos \theta -3\sin \theta }$

39. $r=\frac{3}{\cos \theta -5\sin \theta }$

For the following exercises, find the polar coordinates of the point.

40.

41.

42.

43.

44.

For the following exercises, plot the points.

45. $\left(-2,\frac{\pi }{3}\right)$

46. $\left(-1,-\frac{\pi }{2}\right)$

47. $\left(3.5,\frac{7\pi }{4}\right)$

48. $\left(-4,\frac{\pi }{3}\right)$

49. $\left(5,\frac{\pi }{2}\right)$

50. $\left(4,\frac{-5\pi }{4}\right)$

51. $\left(3,\frac{5\pi }{6}\right)$

52. $\left(-1.5,\frac{7\pi }{6}\right)$

53. $\left(-2,\frac{\pi }{4}\right)$

54. $\left(1,\frac{3\pi }{2}\right)$

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

55. $5x-y=6$

56. $2x+7y=-3$

57. ${x}^{2}+{\left(y - 1\right)}^{2}=1$

58. ${\left(x+2\right)}^{2}+{\left(y+3\right)}^{2}=13$

59. $x=2$

60. ${x}^{2}+{y}^{2}=5y$

61. ${x}^{2}+{y}^{2}=3x$

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

62. $r=6$

63. $r=-4$

64. $\theta =-\frac{2\pi }{3}$

65. $\theta =\frac{\pi }{4}$

66. $r=\sec \theta$

67. $r=-10\sin \theta$

68. $r=3\cos \theta$

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

70. Use a graphing calculator to find the rectangular coordinates of $\left(-3,\frac{3\pi }{7}\right)$. Round to the nearest thousandth.

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

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

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

74. Describe the graph of $r=a\sec \theta ;a>0$.

75. Describe the graph of $r=a\sec \theta ;a<0$.

76. Describe the graph of $r=a\csc \theta ;a>0$.

77. Describe the graph of $r=a\csc \theta ;a<0$.

78. What polar equations will give an oblique line?

For the following exercises, graph the polar inequality.

79. $r<4$

80. $0\le \theta \le \frac{\pi }{4}$

81. $\theta =\frac{\pi }{4},r\ge 2$

82. $\theta =\frac{\pi }{4},r\ge -3$

83. $0\le \theta \le \frac{\pi }{3},r<2$

84. $\frac{-\pi }{6}<\theta \le \frac{\pi }{3},-3<r<2$