Exercises: 10.2 Polar Graphs

1.

1. Graph [latex]r=k\text{,}[/latex] for [latex]k=1, 2, 3\text{.}[/latex] How does the graph change for different values of [latex]k\text{?}[/latex]
2. Write a Cartesian equation for each graph in part (a).

2.

1. Graph [latex]r=k\text{,}[/latex] for [latex]k=-1, -2, -3\text{.}[/latex] How does these graphs compare to the graphs in Problem 1?
2. Write a Cartesian equation for each graph in part (a).

3.

1. Graph [latex]\theta=k\text{,}[/latex] for [latex]k=\dfrac{\pi}{6},~ \dfrac{\pi}{3},~\dfrac{2\pi}{3},~\dfrac{5\pi}{6}\text{.}[/latex] How does the graph change for different values of [latex]k\text{?}[/latex]
2. Write a Cartesian equation for each graph in part (a).

4.

1. Graph [latex]\theta=k\text{,}[/latex] for [latex]k=\dfrac{7\pi}{6},~ \dfrac{5\pi}{3},~\dfrac{7\pi}{3},~\dfrac{11\pi}{6}\text{.}[/latex] How does the graph change for different values of [latex]k\text{?}[/latex]
2. Write a Cartesian equation for each graph in part (a).

5.

Complete the table of values for each equation. Plot the points in order of increasing [latex]\theta\text{.}[/latex] What is different about the two graphs? Equation 1: [latex]~~r=2[/latex]

Equation 2: [latex]~~r=-2[/latex]

6.

Graph each line and label the points with their coordinates. How are the points on the two lines related? Equation 1: [latex]~~\theta = \dfrac{\pi}{4}[/latex]

Equation 1: [latex]~~\theta = \dfrac{5\pi}{4}[/latex]

7.

1. Graph the circle [latex]r=4\cos \theta\text{.}[/latex] Label the points corresponding to [latex]\theta = 0,~\dfrac{\pi}{4},~\dfrac{\pi}{2},~\dfrac{3\pi}{4},[/latex] and [latex]\pi\text{.}[/latex]
2. Complete the table of values. What happens to the graph as [latex]\theta[/latex] increases from [latex]\pi[/latex] to [latex]2\pi\text{?}[/latex]
   

   [latex]\theta[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
   [latex]r[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

3. Find the center and radius of the circle.
4. Give the Cartesian equation of the circle.

8.

1. Graph the circle [latex]r=4\sin \theta\text{.}[/latex] Label the points corresponding to [latex]\theta = 0,~\dfrac{\pi}{4},~\dfrac{\pi}{2},~\dfrac{3\pi}{4},[/latex] and [latex]\pi\text{.}[/latex]
2. Complete the table of values. What happens to the graph as [latex]\theta[/latex] increases from [latex]\pi[/latex] to [latex]2\pi\text{?}[/latex]
   

   [latex]\theta[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
   [latex]r[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

3. Find the center and radius of the circle.
4. Give the Cartesian equation of the circle.

9.

1. Graph [latex]r=2a\sin\theta[/latex] for [latex]a=-2,-1,1,2\text{.}[/latex]
2. How do the graphs change for different values of [latex]a\text{?}[/latex]

10.

1. Graph [latex]r=2a\cos\theta[/latex] for [latex]a=-2,-1,1,2\text{.}[/latex]
2. How do the graphs change for different values of [latex]a\text{?}[/latex]

11.

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

1. [latex]\displaystyle r=1+\sin \theta[/latex]
2. [latex]\displaystyle r=-1+\sin \theta[/latex]
3. [latex]\displaystyle r=1-\sin \theta[/latex]
4. [latex]\displaystyle r=-1-\sin \theta[/latex]

12.

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

1. [latex]\displaystyle r=1+\cos \theta[/latex]
2. [latex]\displaystyle r=-1+\cos \theta[/latex]
3. [latex]\displaystyle r=1-\cos \theta[/latex]
4. [latex]\displaystyle r=-1-\cos \theta[/latex]

13.

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

1. [latex]\displaystyle r=2+\cos \theta[/latex]
2. [latex]\displaystyle r=2-\cos \theta[/latex]
3. [latex]\displaystyle r=1+2\cos \theta[/latex]
4. [latex]\displaystyle r=1-2\cos \theta[/latex]

14.

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

1. [latex]\displaystyle r=2+\sin \theta[/latex]
2. [latex]\displaystyle r=2-\sin \theta[/latex]
3. [latex]\displaystyle r=1+2\sin \theta[/latex]
4. [latex]\displaystyle r=1-2\sin \theta[/latex]

15.

1. Graph the following roses and compare. How is the number of petals related to the value of [latex]n[/latex] in the equation [latex]r=a\sin n\theta\text{?}[/latex]
   [latex]r=\sin 2\theta,~~r=\sin 3\theta,~~r=\sin 4\theta,~~r=\sin 5\theta[/latex]
2. For each graph above, list the values of [latex]\theta[/latex] where the tips of the petals occur.
3. Graph [latex]r=a\sin 3\theta~[/latex] for [latex]a=1,2,[/latex] and [latex]3\text{.}[/latex] How does the value of [latex]a[/latex] affect the graph?

16.

1. Graph the following roses and compare. How is the number of petals related to the value of [latex]n[/latex] in the equation [latex]r=a\cos n\theta\text{?}[/latex]
   [latex]r=\cos 2\theta,~~r=\cos 3\theta,~~r=\cos 4\theta,~~r=\cos 5\theta[/latex]
2. For each graph above, list the values of [latex]\theta[/latex] where the tips of the petals occur.
3. Graph [latex]r=a\cos 3\theta~[/latex] for [latex]a=1,2,[/latex] and [latex]3\text{.}[/latex] How does the value of [latex]a[/latex] affect the graph?

17.

1. Solve [latex]r^2=9\cos 2\theta~[/latex] for [latex]r\text{.}[/latex] (You should get two equations for [latex]r\text{.}[/latex])
2. Graph both equations together. Change [latex]\theta[/latex]step to 0.02 to see the whole graph.
3. How does the value of [latex]a[/latex] affect the graph of [latex]r^2=a^2\cos 2\theta\text{?}[/latex]

18.

1. Solve [latex]r^2=9\cos 2\theta~[/latex] for [latex]r\text{.}[/latex] (You should get two equations for [latex]r\text{.}[/latex])
2. Graph both equations together. Change [latex]\theta[/latex]step to 0.02 to see the whole graph.
3. How does this graph differ from the graph in Problem 17?

19.

Graph the Archimedean spiral [latex]r=\theta\text{.}[/latex] Set your window to

[latex][latex] \begin{aligned}[t] \theta\text{min}=0~~~~~~~~~~\theta\text{max}=8\pi\\ \text{Xmin}=-20~~~~\text{Xmax}=20\\ \text{Ymin}=-20~~~~\text{Ymax}=20\\ \end{aligned}[/latex]

Then graph by pressing Zoom 5.

20.

Graph the logarithmic spiral [latex]r=e^{0.2\theta}\text{.}[/latex] Set your window to

[latex][latex] \begin{aligned}[t] \theta\text{min}=0~~~~~~~~~~\theta\text{max}=8\pi\\ \text{Xmin}=-100~~~~\text{Xmax}=100\\ \text{Ymin}=-100~~~~\text{Ymax}=100\\ \end{aligned}[/latex]

Then graph by pressing Zoom 5.

21.

1. 1. Complete the table and graph the equation [latex]y=\sin 3\theta[/latex] in Cartesian coordinates, for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
      

      [latex]\theta[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]
      [latex]3\theta[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\dfrac{3\pi}{4}[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
      [latex]y[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

Complete the table and graph the equation [latex]r=\sin 3\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]

22.

1. 1. Complete the table and graph the equation [latex]y=\cos 2\theta[/latex] in Cartesian coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
      

      [latex]\theta[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]
      [latex]2\theta[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\dfrac{3\pi}{4}[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
      [latex]y[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

Complete the table and graph the equation [latex]r=\cos 2\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]

23.

1. 1. Complete the table and graph the equation [latex]y=2+2\cos\theta[/latex] in Cartesian coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
      

      [latex]\theta[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\dfrac{3\pi}{4}[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
      [latex]y[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

Complete the table and graph the equation [latex]r=2+2\cos\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]

24.

1. 1. Complete the table and graph the equation [latex]y=1-\sin\theta[/latex] in Cartesian coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
      

      [latex]\theta[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\dfrac{3\pi}{4}[/latex]	[latex]\pi[/latex]	[latex]\dfrac{5\pi}{4}[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]\dfrac{7\pi}{4}[/latex]	[latex]2\pi[/latex]
      [latex]y[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]	[latex]\hphantom{000}[/latex]

Complete the table and graph the equation [latex]r=1-\sin\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]

25.

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

26.

[latex]r=2\sin \theta[/latex]

27.

[latex]\theta=\dfrac{\pi}{4}[/latex]

28.

[latex]\theta=\dfrac{4\pi}{3}[/latex]

29.

[latex]r=4[/latex]

30.

[latex]r=2[/latex]

31.

[latex]r=2+2\sin \theta[/latex]

32.

[latex]r=3+3\cos\theta[/latex]

33.

[latex]r=2-\cos\theta[/latex]

34.

[latex]r=1-3\sin \theta[/latex]

35.

[latex]r=3\sin 2\theta[/latex]

36.

[latex]r=2\cos 3\theta[/latex]

37.

[latex]r=2\cos 5\theta[/latex]

38.

[latex]r=4\sin 4\theta[/latex]

39.

[latex]r=2+3\sin \theta[/latex]

40.

[latex]r=3+2\sin \theta[/latex]

41.

[latex]r^2=\cos 2\theta[/latex]

42.

[latex]r^2=4\sin2\theta[/latex]

43.

[latex]r\csc\theta = 2[/latex]

44.

[latex]r=2\sec \theta[/latex]

45.

[latex]r^2=4,~ 0 \le \theta \le \dfrac{3\pi}{4}[/latex]

46.

[latex]\theta = \dfrac{\pi}{4},~ {|r|} \lt 2[/latex]

47.

[latex]r=\sin \theta,~ \dfrac{3\pi}{4} \le \theta \le \dfrac{5\pi}{4}[/latex]

48.

[latex]r=\cos \theta,~ 0 \le \theta \le \dfrac{\pi}{2}[/latex]

49.

[latex]r=2\sin 2\theta \cos 2\theta[/latex]

50.

[latex]r=\cos^2 \theta - \sin^2 \theta[/latex]

51.

[latex]r(1-\cos\theta)=\sin^2\theta[/latex]

52.

[latex]r \sec \theta=\sec\theta - \tan \theta[/latex]

53.

[latex]r = \dfrac{2}{1-\cos\theta}[/latex]

54.

[latex]r = \dfrac{6}{2+\sin\theta}[/latex]

55.

[latex]r = \dfrac{2}{2-\cos\theta}[/latex]

56.

[latex]r = \dfrac{1}{1+\sin\theta}[/latex]

57.

[latex]r = \dfrac{1}{1+2\sin\theta}[/latex]

58.

[latex]r = \dfrac{3}{2-3\cos\theta}[/latex]

59.

60.

61.

62.

63.

64.

65.

66.

67.

[latex]r=\cos\theta,~ r=1-\cos\theta[/latex]

68.

[latex]r=\sin\theta,~ r=\cos\theta[/latex]

69.

[latex]r=3\sin\theta,~ r=3\cos\theta[/latex]

70.

[latex]r=\sin 2\theta,~ r=\cos 2\theta[/latex]

71.

[latex]r=1,~ r=1-\cos\theta[/latex]

72.

[latex]r=3\cos\theta,~ r=1+\cos\theta[/latex]

73.

[latex]r=2+\sin\theta,~ r=2-\cos\theta[/latex]

74.

[latex]r=\sin\theta,~ r=\sin 2\theta[/latex]

75.

[latex]r^2 = \tan \theta[/latex]

76.

[latex]r^2 = \cot \theta[/latex]

77.

[latex]r=\csc\theta-2[/latex] (conchoid)

78.

[latex]r=\tan\theta[/latex] (kappa curve)

79.

[latex]r=\cos 2\theta \sec\theta[/latex] (strophoid)

80.

[latex]r=\sin \theta \tan \theta[/latex] (cissoid)

81.

[latex]r=\dfrac{1}{\sqrt{\theta}}[/latex]

82.

[latex]r=\cos\ \dfrac{\theta}{2},~ 0 \le \theta \le 4\pi[/latex]

83.

Graph the polar curves [latex]r=1-2\sin n\theta[/latex] for [latex]n=2,3,4,5,6\text{.}[/latex] Explain how the value of the parameter [latex]n[/latex] affects the curve.

84.

Graph the polar curves [latex]r=1-3\cos n\theta[/latex] for [latex]n=2,3,4,5,6\text{.}[/latex] Explain how the value of the parameter [latex]n[/latex] affects the curve.