Exercises: 10.2 Polar Graphs

1.

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

2.

1. Graph $r=k\text{,}$ for $k=-1,-2,-3\text{.}$ 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 $\theta =k\text{,}$ for $k={\displaystyle \frac{\pi }{6}},{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{2\pi }{3}},{\displaystyle \frac{5\pi }{6}}\text{.}$ How does the graph change for different values of $k\text{?}$
2. Write a Cartesian equation for each graph in part (a).

4.

1. Graph $\theta =k\text{,}$ for $k={\displaystyle \frac{7\pi }{6}},{\displaystyle \frac{5\pi }{3}},{\displaystyle \frac{7\pi }{3}},{\displaystyle \frac{11\pi }{6}}\text{.}$ How does the graph change for different values of $k\text{?}$
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 $\theta \text{.}$ What is different about the two graphs? Equation 1: $r=2$

Equation 2: $r=-2$

6.

Graph each line and label the points with their coordinates. How are the points on the two lines related? Equation 1: $\theta ={\displaystyle \frac{\pi }{4}}$

Equation 1: $\theta ={\displaystyle \frac{5\pi }{4}}$

7.

1. Graph the circle $r=4\cos \theta \text{.}$ Label the points corresponding to $\theta =0,{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{4}},$ and $\pi \text{.}$
2. Complete the table of values. What happens to the graph as $\theta$ increases from $\pi$ to $2\pi \text{?}$
   

   $\theta$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
   $r$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

8.

1. Graph the circle $r=4\sin \theta \text{.}$ Label the points corresponding to $\theta =0,{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{4}},$ and $\pi \text{.}$
2. Complete the table of values. What happens to the graph as $\theta$ increases from $\pi$ to $2\pi \text{?}$
   

   $\theta$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
   $r$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

9.

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

10.

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

11.

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

1. ${\displaystyle r=1+\sin \theta }$
2. ${\displaystyle r=-1+\sin \theta }$
3. ${\displaystyle r=1-\sin \theta }$
4. ${\displaystyle r=-1-\sin \theta }$

12.

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

1. ${\displaystyle r=1+\cos \theta }$
2. ${\displaystyle r=-1+\cos \theta }$
3. ${\displaystyle r=1-\cos \theta }$
4. ${\displaystyle r=-1-\cos \theta }$

13.

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

1. ${\displaystyle r=2+\cos \theta }$
2. ${\displaystyle r=2-\cos \theta }$
3. ${\displaystyle r=1+2\cos \theta }$
4. ${\displaystyle r=1-2\cos \theta }$

14.

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

1. ${\displaystyle r=2+\sin \theta }$
2. ${\displaystyle r=2-\sin \theta }$
3. ${\displaystyle r=1+2\sin \theta }$
4. ${\displaystyle r=1-2\sin \theta }$

15.

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

16.

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

17.

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

18.

1. Solve $r^2=9\cos 2\theta$ for $r\text{.}$ (You should get two equations for $r\text{.}$)
2. Graph both equations together. Change $\theta$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 $r=\theta \text{.}$ Set your window to

$[latex]\begin{array}{r}\theta \text{min}=0\theta \text{max}=8\pi \\ \text{Xmin}=-20\text{Xmax}=20\\ \text{Ymin}=-20\text{Ymax}=20\end{array}$

Then graph by pressing Zoom 5.

20.

Graph the logarithmic spiral $r=e^{0.2\theta }\text{.}$ Set your window to

$[latex]\begin{array}{r}\theta \text{min}=0\theta \text{max}=8\pi \\ \text{Xmin}=-100\text{Xmax}=100\\ \text{Ymin}=-100\text{Ymax}=100\end{array}$

Then graph by pressing Zoom 5.

21.

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

      $\theta$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$
      $3\theta$	$0$	${\displaystyle \frac{\pi }{4}}$	${\displaystyle \frac{\pi }{2}}$	${\displaystyle \frac{3\pi }{4}}$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
      $y$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

22.

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

      $\theta$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$
      $2\theta$	$0$	${\displaystyle \frac{\pi }{4}}$	${\displaystyle \frac{\pi }{2}}$	${\displaystyle \frac{3\pi }{4}}$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
      $y$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

23.

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

      $\theta$	$0$	${\displaystyle \frac{\pi }{4}}$	${\displaystyle \frac{\pi }{2}}$	${\displaystyle \frac{3\pi }{4}}$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
      $y$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

24.

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

      $\theta$	$0$	${\displaystyle \frac{\pi }{4}}$	${\displaystyle \frac{\pi }{2}}$	${\displaystyle \frac{3\pi }{4}}$	$\pi$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{7\pi }{4}}$	$2\pi$
      $y$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$	$\phantom{000}$

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

25.

$r=3\cos \theta$

26.

$r=2\sin \theta$

27.

$\theta ={\displaystyle \frac{\pi }{4}}$

28.

$\theta ={\displaystyle \frac{4\pi }{3}}$

29.

$r=4$

30.

$r=2$

31.

$r=2+2\sin \theta$

32.

$r=3+3\cos \theta$

33.

$r=2-\cos \theta$

34.

$r=1-3\sin \theta$

35.

$r=3\sin 2\theta$

36.

$r=2\cos 3\theta$

37.

$r=2\cos 5\theta$

38.

$r=4\sin 4\theta$

39.

$r=2+3\sin \theta$

40.

$r=3+2\sin \theta$

41.

$r^2=\cos 2\theta$

42.

$r^2=4\sin 2\theta$

43.

$r\csc \theta =2$

44.

$r=2\sec \theta$

45.

$r^2=4,0\le \theta \le {\displaystyle \frac{3\pi }{4}}$

46.

$\theta ={\displaystyle \frac{\pi }{4}},|r|<2$

47.

$r=\sin \theta ,{\displaystyle \frac{3\pi }{4}}\le \theta \le {\displaystyle \frac{5\pi }{4}}$

48.

$r=\cos \theta ,0\le \theta \le {\displaystyle \frac{\pi }{2}}$

49.

$r=2\sin 2\theta \cos 2\theta$

50.

$r={\cos }^2\theta -{\sin }^2\theta$

51.

$r(1-\cos \theta )={\sin }^2\theta$

52.

$r\sec \theta =\sec \theta -\tan \theta$

53.

$r={\displaystyle \frac{2}{1-\cos \theta }}$

54.

$r={\displaystyle \frac{6}{2+\sin \theta }}$

55.

$r={\displaystyle \frac{2}{2-\cos \theta }}$

56.

$r={\displaystyle \frac{1}{1+\sin \theta }}$

57.

$r={\displaystyle \frac{1}{1+2\sin \theta }}$

58.

$r={\displaystyle \frac{3}{2-3\cos \theta }}$

59.

60.

61.

62.

63.

64.

65.

66.

67.

$r=\cos \theta ,r=1-\cos \theta$

68.

$r=\sin \theta ,r=\cos \theta$

69.

$r=3\sin \theta ,r=3\cos \theta$

70.

$r=\sin 2\theta ,r=\cos 2\theta$

71.

$r=1,r=1-\cos \theta$

72.

$r=3\cos \theta ,r=1+\cos \theta$

73.

$r=2+\sin \theta ,r=2-\cos \theta$

74.

$r=\sin \theta ,r=\sin 2\theta$

75.

$r^2=\tan \theta$

76.

$r^2=\cot \theta$

77.

$r=\csc \theta -2$ (conchoid)

78.

$r=\tan \theta$ (kappa curve)

79.

$r=\cos 2\theta \sec \theta$ (strophoid)

80.

$r=\sin \theta \tan \theta$ (cissoid)

81.

$r={\displaystyle \frac{1}{\sqrt{\theta }}}$

82.

$r=\cos {\displaystyle \frac{\theta }{2}},0\le \theta \le 4\pi$

83.

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

84.

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