In the following exercises, plot the point whose polar coordinates are given by first constructing the angle [latex]\theta [/latex] and then marking off the distance r along the ray.
1. [latex]\left(3,\frac{\pi }{6}\right)[/latex]
2. [latex]\left(-2,\frac{5\pi }{3}\right)[/latex]
3. [latex]\left(0,\frac{7\pi }{6}\right)[/latex]
4. [latex]\left(-4,\frac{3\pi }{4}\right)[/latex]
5. [latex]\left(1,\frac{\pi }{4}\right)[/latex]
6. [latex]\left(2,\frac{5\pi }{6}\right)[/latex]
7. [latex]\left(1,\frac{\pi }{2}\right)[/latex]
For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.
8. Coordinates of point A.
9. Coordinates of point B.
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[latex]B\begin{array}{cc}\left(3,\frac{\text{-}\pi }{3}\right)\hfill & B\left(-3,\frac{2\pi }{3}\right)\hfill \end{array}[/latex]
10. Coordinates of point C.
11. Coordinates of point D.
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[latex]D\left(5,\frac{7\pi }{6}\right)D\left(-5,\frac{\pi }{6}\right)[/latex]
For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\left(0,2\pi \right][/latex]. Round to three decimal places.
12. [latex]\left(2,2\right)[/latex]
13. [latex]\left(3,-4\right)[/latex]
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[latex]\begin{array}{cc}\left(5,-0.927\right)\hfill & \left(-5,-0.927+\pi \right)\hfill \end{array}[/latex]
14. [latex]\left(8,15\right)[/latex]
15. [latex]\left(-6,8\right)[/latex]
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[latex]\left(10,-0.927\right)\left(-10,-0.927+\pi \right)[/latex]
16. [latex]\left(4,3\right)[/latex]
17. [latex]\left(3,\text{-}\sqrt{3}\right)[/latex]
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[latex]\left(2\sqrt{3},-0.524\right)\left(-2\sqrt{3},-0.524+\pi \right)[/latex]
For the following exercises, find rectangular coordinates for the given point in polar coordinates.
18. [latex]\left(2,\frac{5\pi }{4}\right)[/latex]
19. [latex]\left(-2,\frac{\pi }{6}\right)[/latex]
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[latex]\left(\begin{array}{cc}\text{-}\sqrt{3},\hfill & -1\hfill \end{array}\right)[/latex]
20. [latex]\left(5,\frac{\pi }{3}\right)[/latex]
21. [latex]\left(1,\frac{7\pi }{6}\right)[/latex]
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[latex]\left(\begin{array}{cc}-\frac{\sqrt{3}}{2},\hfill & \frac{-1}{2}\hfill \end{array}\right)[/latex]
22. [latex]\left(-3,\frac{3\pi }{4}\right)[/latex]
23. [latex]\left(0,\frac{\pi }{2}\right)[/latex]
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[latex]\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)[/latex]
24. [latex]\left(-4.5,6.5\right)[/latex]
For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[/latex] -axis, the [latex]y[/latex] -axis, or the origin.
25. [latex]r=3\sin\left(2\theta \right)[/latex]
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Symmetry with respect to the x-axis, y-axis, and origin.
26. [latex]{r}^{2}=9\cos\theta [/latex]
27. [latex]r=\cos\left(\frac{\theta }{5}\right)[/latex]
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Symmetric with respect to x-axis only.
28. [latex]r=2\sec\theta [/latex]
29. [latex]r=1+\cos\theta [/latex]
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Symmetry with respect to x-axis only.
For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.
31. [latex]\theta =\frac{\pi }{4}[/latex]
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Line [latex]y=x[/latex]
32. [latex]r=\sec\theta [/latex]
33. [latex]r=\csc\theta [/latex]
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[latex]y=1[/latex]
For the following exercises, convert the rectangular equation to polar form and sketch its graph.
34. [latex]{x}^{2}+{y}^{2}=16[/latex]
35. [latex]{x}^{2}-{y}^{2}=16[/latex]
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Hyperbola; polar form [latex]{r}^{2}\cos\left(2\theta \right)=16[/latex] or [latex]{r}^{2}=16\sec\theta [/latex].
For the following exercises, convert the rectangular equation to polar form and sketch its graph.
37. [latex]3x-y=2[/latex]
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[latex]r=\frac{2}{3\cos\theta -\sin\theta }[/latex]
38. [latex]{y}^{2}=4x[/latex]
For the following exercises, convert the polar equation to rectangular form and sketch its graph.
39. [latex]r=4\sin\theta [/latex]
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[latex]{x}^{2}+{y}^{2}=4y[/latex]
40. [latex]r=6\cos\theta [/latex]
41. [latex]r=\theta [/latex]
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[latex]x\tan\sqrt{{x}^{2}+{y}^{2}}=y[/latex]
42. [latex]r=\cot\theta \csc\theta [/latex]
For the following exercises, sketch a graph of the polar equation and identify any symmetry.
43. [latex]r=1+\sin\theta [/latex]
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y-axis symmetry
44. [latex]r=3 - 2\cos\theta [/latex]
45. [latex]r=2 - 2\sin\theta [/latex]
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y-axis symmetry
46. [latex]r=5 - 4\sin\theta [/latex]
47. [latex]r=3\cos\left(2\theta \right)[/latex]
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x- and y-axis symmetry and symmetry about the pole
48. [latex]r=3\sin\left(2\theta \right)[/latex]
49. [latex]r=2\cos\left(3\theta \right)[/latex]
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x-axis symmetry
50. [latex]r=3\cos\left(\frac{\theta }{2}\right)[/latex]
51. [latex]{r}^{2}=4\cos\left(2\theta \right)[/latex]
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x- and y-axis symmetry and symmetry about the pole
52. [latex]{r}^{2}=4\sin\theta [/latex]
53. [latex]r=2\theta [/latex]
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no symmetry
54. [T] The graph of [latex]r=2\cos\left(2\theta \right)\sec\left(\theta \right)[/latex]. is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.
55. [T] Use a graphing utility and sketch the graph of [latex]r=\frac{6}{2\sin\theta -3\cos\theta }[/latex].
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a line
56. [T] Use a graphing utility to graph [latex]r=\frac{1}{1-\cos\theta }[/latex].
57. [T] Use technology to graph [latex]r={e}^{\sin\left(\theta \right)}-2\cos\left(4\theta \right)[/latex].
58. [T] Use technology to plot [latex]r=\sin\left(\frac{3\theta }{7}\right)[/latex] (use the interval [latex]0\le \theta \le 14\pi [/latex]).
59. Without using technology, sketch the polar curve [latex]\theta =\frac{2\pi }{3}[/latex].
60. [T] Use a graphing utility to plot [latex]r=\theta \sin\theta [/latex] for [latex]\text{-}\pi \le \theta \le \pi [/latex].
61. [T] Use technology to plot [latex]r={e}^{-0.1\theta }[/latex] for [latex]-10\le \theta \le 10[/latex].
62. [T] There is a curve known as the “Black Hole.” Use technology to plot [latex]r={e}^{-0.01\theta }[/latex] for [latex]-100\le \theta \le 100[/latex].
63. [T] Use the results of the preceding two problems to explore the graphs of [latex]r={e}^{-0.001\theta }[/latex] and [latex]r={e}^{-0.0001\theta }[/latex] for [latex]|\theta |>100[/latex].
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Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.
