1. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.
2. Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x-axis, y-axis, and origin?
3. What are the steps to follow when graphing polar equations?
4. Describe the shapes of the graphs of cardioids, limaçons, and lemniscates.
5. What part of the equation determines the shape of the graph of a polar equation?
For the following exercises, test the equation for symmetry.
6. [latex]r=5\cos 3\theta [/latex]
7. [latex]r=3 - 3\cos \theta [/latex]
8. [latex]r=3+2\sin \theta [/latex]
9. [latex]r=3\sin 2\theta [/latex]
10. [latex]r=4[/latex]
11. [latex]r=2\theta [/latex]
12. [latex]r=4\cos \frac{\theta }{2}[/latex]
13. [latex]r=\frac{2}{\theta }[/latex]
14. [latex]r=3\sqrt{1-{\cos }^{2}\theta }[/latex]
15. [latex]r=\sqrt{5\sin 2\theta }[/latex]
For the following exercises, graph the polar equation. Identify the name of the shape.
16. [latex]r=3\cos \theta [/latex]
17. [latex]r=4\sin \theta [/latex]
18. [latex]r=2+2\cos \theta [/latex]
19. [latex]r=2 - 2\cos \theta [/latex]
20. [latex]r=5 - 5\sin \theta [/latex]
21. [latex]r=3+3\sin \theta [/latex]
22. [latex]r=3+2\sin \theta [/latex]
23. [latex]r=7+4\sin \theta [/latex]
24. [latex]r=4+3\cos \theta [/latex]
25. [latex]r=5+4\cos \theta [/latex]
26. [latex]r=10+9\cos \theta [/latex]
27. [latex]r=1+3\sin \theta [/latex]
28. [latex]r=2+5\sin \theta [/latex]
29. [latex]r=5+7\sin \theta [/latex]
30. [latex]r=2+4\cos \theta [/latex]
31. [latex]r=5+6\cos \theta [/latex]
32. [latex]{r}^{2}=36\cos \left(2\theta \right)[/latex]
33. [latex]{r}^{2}=10\cos \left(2\theta \right)[/latex]
34. [latex]{r}^{2}=4\sin \left(2\theta \right)[/latex]
35. [latex]{r}^{2}=10\sin \left(2\theta \right)[/latex]
36. [latex]r=3\text{sin}\left(2\theta \right)[/latex]
37. [latex]r=3\text{cos}\left(2\theta \right)[/latex]
38. [latex]r=5\text{sin}\left(3\theta \right)[/latex]
39. [latex]r=4\text{sin}\left(4\theta \right)[/latex]
40. [latex]r=4\text{sin}\left(5\theta \right)[/latex]
41. [latex]r=-\theta [/latex]
42. [latex]r=2\theta [/latex]
43. [latex]r=-3\theta [/latex]
For the following exercises, use a graphing calculator to sketch the graph of the polar equation.
44. [latex]r=\frac{1}{\theta }[/latex]
45. [latex]r=\frac{1}{\sqrt{\theta }}[/latex]
46. [latex]r=2\sin \theta \tan \theta [/latex], a cissoid
47. [latex]r=2\sqrt{1-{\sin }^{2}\theta }[/latex] , a hippopede
48. [latex]r=5+\cos \left(4\theta \right)[/latex]
49. [latex]r=2-\sin \left(2\theta \right)[/latex]
50. [latex]r={\theta }^{2}[/latex]
51. [latex]r=\theta +1[/latex]
52. [latex]r=\theta \sin \theta [/latex]
53. [latex]r=\theta \cos \theta [/latex]
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of [latex]\left[0,4\pi \right][/latex] and then explain the differences shown in the graphs.
54. [latex]r=\theta ,r=-\theta [/latex]
55. [latex]r=\theta ,r=\theta +\sin \theta [/latex]
56. [latex]r=\sin \theta +\theta ,r=\sin \theta -\theta [/latex]
57. [latex]r=2\sin \left(\frac{\theta }{2}\right),r=\theta \sin \left(\frac{\theta }{2}\right)[/latex]
58. [latex]r=\sin \left(\cos \left(3\theta \right)\right)r=\sin \left(3\theta \right)[/latex]
59. On a graphing utility, graph [latex]r=\sin \left(\frac{16}{5}\theta \right)[/latex] on [latex]\left[0,4\pi \right],\left[0,8\pi \right],\left[0,12\pi \right][/latex], and [latex]\left[0,16\pi \right][/latex]. Describe the effect of increasing the width of the domain.
60. On a graphing utility, graph and sketch [latex]r=\sin \theta +{\left(\sin \left(\frac{5}{2}\theta \right)\right)}^{3}[/latex] on [latex]\left[0,4\pi \right][/latex].
61. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.
[latex]\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3\sin \left(3\theta \right)\end{array}\hfill \\ {r}_{2}=2\sin \left(3\theta \right)\hfill \\ {r}_{3}=\sin \left(3\theta \right)\hfill \end{array}[/latex]
62. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.
[latex]\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3+3\cos \theta \end{array}\hfill \\ {r}_{2}=2+2\cos \theta \hfill \\ {r}_{3}=1+\cos \theta \hfill \end{array}[/latex]
63. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.
[latex]\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3\theta \end{array}\hfill \\ {r}_{2}=2\theta \hfill \\ {r}_{3}=\theta \hfill \end{array}[/latex]
For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.
64. [latex]{r}_{1}=3+2\sin \theta ,{r}_{2}=2[/latex]
65. [latex]{r}_{1}=6 - 4\cos \theta ,{r}_{2}=4[/latex]
66. [latex]{r}_{1}=1+\sin \theta ,{r}_{2}=3\sin \theta [/latex]
67. [latex]{r}_{1}=1+\cos \theta ,{r}_{2}=3\cos \theta [/latex]
68. [latex]{r}_{1}=\cos \left(2\theta \right),{r}_{2}=\sin \left(2\theta \right)[/latex]
69. [latex]{r}_{1}={\sin }^{2}\left(2\theta \right),{r}_{2}=1-\cos \left(4\theta \right)[/latex]
70. [latex]{r}_{1}=\sqrt{3},{r}_{2}=2\sin \left(\theta \right)[/latex]
71. [latex]{r}_{1}{}^{2}=\sin \theta ,{r}_{2}{}^{2}=\cos \theta [/latex]
72. [latex]{r}_{1}=1+\cos \theta ,{r}_{2}=1-\sin \theta [/latex]