9.2.2: Homework

Concept Check

1. Mike says that the graph of \(r=2 \sin\left( \theta \right)\) in the Example \( \PageIndex{ 1 } \) cannot be correct because there are no points on the graph for angles between \(\pi\) and \(2 \pi\). How do you respond?

2. Is it possible to have a rose with only two petals? What would its equation be?

3. Linda says that a circle of the form \(r=2 a \cos\left( \theta \right)\) is a special case of a limaçon. Support or refute her statement.

4. There are no points on the graph of \(r^2=4 \cos\left( 2 \theta \right)\) for angles between \(\frac{\pi}{4}\) and \(\frac{3 \pi}{4}\) Why is that?

Basic Skills

For the following exercises, use your calculator to graph the equation.

5.    

   1. Graph \(r=k\), for \(k=1,2,3\). How do these graphs change for different values of \(k\)?

   2. Write a Cartesian equation for each graph in part (a).

6.    

   1. Graph \(r=k\), for \(k=-1,-2,-3\). How do these graphs compare to the graphs in Problem 5?

   2. Write a Cartesian equation for each graph in part (a).

7.    

   1. Graph \(\theta=k\), for \(k=\dfrac{\pi}{6}, \dfrac{\pi}{3}, \dfrac{2 \pi}{3}, \dfrac{5 \pi}{6}\). How do the graphs change for different values of \(k\)?

   2. Write a Cartesian equation for each graph in part (a).

8.    

   1. Graph \(\theta=k\), for \(k=\dfrac{7 \pi}{6}, \dfrac{5 \pi}{3}, \dfrac{7 \pi}{3}, \dfrac{11 \pi}{6}\). How do the graphs change for different values of \(k\)?

   2. Write a Cartesian equation for each graph in part (a).

9. Complete the table of values for each equation. Plot the points in order of increasing \(\theta\). What is different about the two graphs?

   1. \(r = 2\)

      \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)
      \(r=2\)

   2. \(r = −2\)

      \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)
      \(r=-2\)

10. Graph each line and label the points with their coordinates. How are the points on the two lines related?

    1. \(\theta = \dfrac{\pi}{4}\)

       \(\theta = \dfrac{\pi}{4}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{4}\)
       \(r\)	-2	-1	0	1	2

    2. \(\theta = \dfrac{5\pi}{4}\)

       \(\theta = \dfrac{5\pi}{4}\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{5\pi}{4}\)
       \(r\)	-2	-1	0	1	2

11.  

    1. Graph the circle \(r = 4 \cos\left( \theta \right)\). Label the points corresponding to \(\theta = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}\), and \(\pi\).

    2. Complete the table of values. What happens to the graph as \(\theta\) increases from \(\pi\) to \(2\pi\)?

       \(\theta\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

    3. Find the center and radius of the circle.

    4. Give the Cartesian equation of the circle.

12.  

    1. Graph the circle \(r = 4 \sin\left( \theta \right)\). Label the points corresponding to \(\theta = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}\), and \(\pi\).

    2. Complete the table of values. What happens to the graph as \(\theta\) increases from \(\pi\) to \(2\pi\)?

       \(\theta\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

    3. Find the center and radius of the circle.

    4. Give the Cartesian equation of the circle.

13.  

    1. Graph \(r=2 a \sin\left( \theta \right)\) for \(a=-2,-1,1,2\).

    2. How do the graphs change for different values of \(a\)?

14.    

    1. Graph \(r=2 a \cos\left( \theta \right)\) for \(a=-2,-1,1,2\).

    2. How do the graphs change for different values of \(a\)?

15. Complete the table of values for each cardioid and graph the equation.

    \(\theta\)	0	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
    \(r\)

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

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

    3. \(r=1-\sin\left( \theta \right)\)

    4. \(r=-1-\sin\left( \theta \right)\)

16. Complete the table of values for each cardioid and graph the equation.

    \(\theta\)	0	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
    \(r\)

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

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

    3. c \(r=1-\cos\left( \theta \right)\)

    4. \(r=-1-\cos\left( \theta \right)\)

17. Complete the table of values for each limaçon and graph the equation.

    \(\theta\)	0	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
    \(r\)

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

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

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

    4. \(r=1-2 \cos\left( \theta \right)\)

18. Complete the table of values for each limaçon and graph the equation.

    \(\theta\)	0	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
    \(r\)

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

    2. \(r=2-\sin\left( \theta \right)\)

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

    4. \(r=1-2 \sin\left( \theta \right)\)

19.  

    1. Graph the following roses and compare. How is the number of petals related to the value of \(n\) in the equation \(r=a \sin n \theta\)?\[ r=\sin\left( 2 \theta \right), \quad r=\sin\left( 3 \theta \right), \quad r=\sin\left( 4 \theta \right), \quad r=\sin\left( 5 \theta \right)\) \nonumber \]

    2. For each graph above, list the values of \(\theta\) where the tips of the petals occur.

    3. Graph \(r=a \sin\left( 3 \theta \right)\) for \(a=1\), \(2\), and \(3\). How does the value of \(a\) affect the graph?

20.  

    1. Graph the following roses and compare. How is the number of petals related to the value of \(n\) in the equation \(r=a \cos n \theta\)?\[ r=\cos\left( 2 \theta \right), \quad r=\cos\left( 3 \theta \right), \quad r=\cos\left( 4 \theta \right), \quad r=\cos\left( 5 \theta \right) \nonumber \]

    2. For each graph above, list the values of \(\theta\) where the tips of the petals occur.

    3. Graph \(r=a \cos\left( 3 \theta \right)\) for \(a=1,2\), and 3 . How does the value of \(a\) affect the graph?

21.    

    1. Solve \(r^2=9 \cos\left( 2 \theta \right)\) for \(r\). (You should get two equations for \(r\).)

    2. Graph both equations together. Change the \(\theta\) step to 0.02 to see the whole graph.

    3. How does the value of \(a\) affect the graph of \(r^2=a^2 \cos\left( 2 \theta \right)\)?

22.    

    1. Solve \(r^2=9 \cos\left( 2 \theta \right)\) for \(r\). (You should get two equations for \(r\).)

    2. Graph both equations together. Change the \(\theta\) step to 0.02 to see the whole graph.

    3. How does this graph differ from the graph in Problem 21?

23. Graph the Archimedean spiral \(r=\theta\). Set your window to\[ \begin{array}{cc}
    \theta \min =0 & \theta \max =8 \pi \\[6pt] \mathrm{Xmin}=-20 & \mathrm{X} \max =20 \\[6pt] \mathrm{Y} \min =-20 & \mathrm{Y} \max =20 \\[6pt] \end{array}\nonumber \]Then graph by pressing Zoom 5.

24. Graph the logarithmic spiral \(r=e^{0.2 \theta}\). Set your window to\[ \begin{array}{cc}
    \theta \min =0 & \theta \max =8 \pi \\[6pt] \mathrm{Xmin}=-100 & \mathrm{X} \max =100 \\[6pt] \mathrm{Ymin}=-100 & \mathrm{Y} \max =100 \\[6pt] \end{array}\nonumber \]Then graph by pressing Zoom 5.

25.  

    1. Complete the table and graph the equation \(y=\sin\left( 3 \theta \right)\) in Cartesian coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)
       \(3\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(y\)

    2. Complete the table and graph the equation \(r = \sin\left( 3 \theta \right)\) in polar coordinates, for \(0 \leq \theta \leq 2\pi\).

       \(\theta\)
       \(3\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

26.  

    1. Complete the table and graph the equation \(y=\cos\left( 2 \theta \right)\) in Cartesian coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)
       \(2\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(y\)

    2. Complete the table and graph the equation \(r=\cos\left( 2 \theta \right)\) in polar coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)
       \(2\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

27.  

    1. Complete the table and graph the equation \(y=2+2 \cos\left( \theta \right)\) in Cartesian coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(y\)

    2. Complete the table and graph the equation \(r=2+2 \cos\left( \theta \right)\) in polar coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

28.  

    1. Complete the table and graph the equation \(y=1-\sin\left( \theta \right)\) in Cartesian coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(y\)

    2. Complete the table and graph the equation \(r=1-\sin\left( \theta \right)\) in polar coordinates, for \(0 \leq \theta \leq 2 \pi\).

       \(\theta\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)	\(\dfrac{5\pi}{4}\)	\(\dfrac{3\pi}{2}\)	\(\dfrac{7\pi}{4}\)	\(2\pi\)
       \(r\)

For the following exercises, use the catalog of polar graphs to help you identify and sketch the curve. Check your work by graphing with a calculator.

29. \(r=3 \cos\left( \theta \right)\)

30. \(r=2 \sin\left( \theta \right)\)

31. \(\theta=\dfrac{\pi}{4}\)

32. \(\theta=\dfrac{4 \pi}{3}\)

33. \(r=4\)

34. \(r=2\)

35. \(r=2+2 \sin\left( \theta \right)\)

36. \(r=3+3 \cos\left( \theta \right)\)

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

38. \(r=1-3 \sin\left( \theta \right)\)

39. \(r=3 \sin\left( 2 \theta \right)\)

40. \(r=2 \cos\left( 3 \theta \right)\)

41. \(r=2 \cos\left( 5 \theta \right)\)

42. \(r=4 \sin\left( 4 \theta \right)\)

43. \(r=2+3 \sin\left( \theta \right)\)

44. \(r=3+2 \sin\left( \theta \right)\)

45. \(r^2=\cos\left( 2 \theta \right)\)

46. \(r^2=4 \sin\left( 2 \theta \right)\)

For the following exercises, identify the curve and graph it.

47. \(r \csc \theta=2\)

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

49. \(r^2=4,0 \leq \theta \leq \dfrac{3 \pi}{4}\)

50. \(\theta=\dfrac{\pi}{4},|r|<2\)

51. \(r=\sin\left( \theta \right), \dfrac{3 \pi}{4} \leq \theta \leq \dfrac{5 \pi}{4}\)

52. \(r=\cos\left( \theta \right), 0 \leq \theta \leq \dfrac{\pi}{2}\)

53. \(r=2 \sin\left( 2 \theta \right) \cos\left( 2 \theta \right)\)

54. \(r=\cos ^2 \theta-\sin ^2 \theta\)

55. \(r(1-\cos\left( \theta \right))=\sin ^2 \theta\)

56. \(r \sec \theta=\sec \theta-\tan\left( \theta \right)\)

For the following exercises, graph the polar curve. Do you recognize it?

57. \(r=\dfrac{2}{1-\cos\left( \theta \right)}\)

58. \(r=\dfrac{6}{2+\sin\left( \theta \right)}\)

59. \(r=\dfrac{2}{2-\cos\left( \theta \right)}\)

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

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

62. \(r=\dfrac{3}{2-3 \cos\left( \theta \right)}\)

For the following exercises, write a polar equation for the graph.

For the following exercises, find the coordinates of the intersection points of the two curves analytically. Then, graph the curves to verify your answers.

71. \(r=\cos\left( \theta \right), r=1-\cos\left( \theta \right)\)

72. \(r=\sin\left( \theta \right), r=\cos\left( \theta \right)\)

73. \(r=3 \sin\left( \theta \right), r=3 \cos\left( \theta \right)\)

74. \(r=\sin\left( 2 \theta \right), r=\cos\left( 2 \theta \right)\)

75. \(r=1, r=1-\cos\left( \theta \right)\)

76. \(r=3 \cos\left( \theta \right), r=1+\cos\left( \theta \right)\)

77. \(r=2+\sin\left( \theta \right), r=2-\cos\left( \theta \right)\)

78. \(r=\sin\left( \theta \right), r=\sin\left( 2 \theta \right)\)

For the following exercises, graph the polar curve.

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

80. \(r^2=\cot \theta\)

81. \(r=\csc \theta-2\) (conchoid)

82. \(r=\tan\left( \theta \right)\) (kappa curve)

83. \(r=\cos\left( 2 \theta \right) \sec \theta\) (strophoid)

84. \(r=\sin\left( \theta \right) \tan\left( \theta \right)\) (cissoid)

85. \(r=\dfrac{1}{\sqrt{\theta}}\)

86. \(r=\cos \dfrac{\theta}{2}, 0 \leq \theta \leq 4 \pi\)

87. Graph the polar curves \(r=1-2 \sin n \theta\) for \(n=2,3,4,5,6\). Explain how the value of the parameter \(n\) affects the curve.

88. Graph the polar curves \(r=1-3 \cos n \theta\) for \(n=2,3,4,5,6\). Explain how the value of the parameter \(n\) affects the curve.