9.1.2: Homework

Vocabulary Check

1. The origin in the polar coordinate system is called the ___.

2. The positive \( x \)-axis in the polar coordinate system is called the ___.

Concept Check

3. Why is the Cartesian coordinate system sometimes called the rectangular system?

4. What do the graphs of the polar equations \(r=k\) and \(\theta=k\), where \(k\) is a constant, look like?

5. Fill in the blank: the point with polar coordinates \((-6, \theta)\) also has coordinates \((6, ?)\).

6. What does the region described by \(a \leq r \leq b\) look like? What does the region described by \(a \leq \theta \leq b\) look like?

7. Sketch a figure that explains how to convert between polar and Cartesian coordinates.

Basic Skills

For the following exercises, find exact values for the coordinates of the point.

For the following exercises, find the coordinates of the point, rounded to hundredths.

For the following exercises, use the grid below to plot the points whose polar coordinates are given.

20. \(\left(2, \dfrac{2 \pi}{3}\right)\)

21. \(\left(3, \dfrac{5 \pi}{4}\right)\)

22. \(\left(3, \dfrac{3 \pi}{2}\right)\)

23. \((4, \pi)\)

24. \(\left(-4, \dfrac{\pi}{6}\right)\)

25. \(\left(-1, \dfrac{\pi}{2}\right)\)

26. \(\left(-1, \dfrac{7 \pi}{4}\right)\)

27. \(\left(-2, \dfrac{5 \pi}{3}\right)\)

For the following exercises, give polar coordinates for each point shown below, with \(r \geq 0\) and \(0 \leq \theta \leq 2 \pi\).

28. A

29. B

30. C

31. D

32. E

33. F

34. G

35. H

For the following exercises, convert the polar coordinates to Cartesian coordinates.

36. \(\left(6, \dfrac{2 \pi}{3}\right)\)

37. \(\left(5, \dfrac{7 \pi}{6}\right)\)

38. \(\left(-3, \dfrac{3 \pi}{4}\right)\)

39. \(\left(-4, \dfrac{5 \pi}{3}\right)\)

40. \((2.4,3.6)\)

41. \((1.7,5.2)\)

42. \((-2,1.5)\)

43. \((-3,4.5)\)

For the following exercises, convert the Cartesian coordinates to polar coordinates with \(r \geq 0\) and \(0 \leq \theta \leq 2 \pi\). Give exact values for \(r\) and \(\theta\).

44. \((7,7)\)

45. \((-7,-7)\)

46. \((\sqrt{6},-\sqrt{2})\)

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

48. \((-3,-2)\)

49. \((1,4)\)

50. \((-2,0)\)

51. \((0,-5)\)

For the following exercises, polar coordinates are given.
(a) Convert to polar coordinates with \(r \leq 0,0 \leq \theta<2 \pi\).
(b) Convert to polar coordinates with \(r \geq 0, \quad-2 \pi<\theta \leq 0\).

52. \(\left(2, \dfrac{5 \pi}{6}\right)\)

53. \(\left(1, \dfrac{4 \pi}{3}\right)\)

54. \((3, \pi)\)

55. \((4,0)\)

56. \((2.3,5.2)\)

57. \((1.2,1.2)\)

For the following exercises, sketch the region described by the inequalities.

58. \(r \leq 3\)

59. \(2 \leq r \leq 3\)

60. \(\dfrac{\pi}{6} \leq \theta \leq \dfrac{\pi}{3}\)

61. \(0 \leq \theta \leq \dfrac{\pi}{2}\)

62. \(r \geq 4, \dfrac{\pi}{2} \leq \theta \leq \dfrac{3 \pi}{4}\)

63. \(2 \leq r \leq 3, \dfrac{5 \pi}{3} \leq \theta \leq \dfrac{11 \pi}{6}\)

For the following exercises, write inequalities to describe the region.

For the following exercises, convert the equation into Cartesian coordinates.

70. \(r^2=2\)

71. \(r=2\)

72. \(r=4 \cos\left( \theta \right)\)

73. \(r=\sin\left( \theta \right)\)

74. \(r=\csc\left( \theta \right)\)

75. \(r=2 \sec\left( \theta \right)\)

76. \(\tan\left( \theta \right)=2\)

77. \(r^2=\tan\left( \theta \right)\)

78. \(r \sec\left( \theta \right)=3\)

79. \(\theta=\dfrac{\pi}{3}\)

80. \(r=\dfrac{2}{1+\sin\left( \theta \right)}\)

81. \(r=\dfrac{4}{4-\cos\left( \theta \right)}\)

82. \(2 r \cos\left( \theta \right)+r \sin\left( \theta \right)=1\)

83. \(\tan\left( \theta \right)=r \cos\left( \theta \right)-2\)

For the following exercises, convert the equation into polar coordinates.

84. \(x=2\)

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

86. \(2 x y=1\)

87. \(y=-x\)

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

89. \(x^2-y^2=1\)

90. \(x^2+y^2=4 \sqrt{x^2+y^2}\)

91. \(\left(x^2+y^2\right)^3=9 x^2 y^2\)

Synthesis Questions

92. Show that the graph of \(r=a \cos\left( \theta \right)+b \sin\left( \theta \right)\) is a circle. Find its center and radius.

Proofs

93. Use the Law of Cosines to prove the Distance Formula in polar coordinates:\[d=\sqrt{r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_2-\theta_1\right)}\nonumber \]