Problem Set: Polar Coordinates

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.

The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.

8. Coordinates of point A.

9. Coordinates of point B.

10. Coordinates of point C.

11. Coordinates of point D.

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]

14. [latex]\left(8,15\right)[/latex]

15. [latex]\left(-6,8\right)[/latex]

16. [latex]\left(4,3\right)[/latex]

17. [latex]\left(3,\text{-}\sqrt{3}\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]

20. [latex]\left(5,\frac{\pi }{3}\right)[/latex]

21. [latex]\left(1,\frac{7\pi }{6}\right)[/latex]

22. [latex]\left(-3,\frac{3\pi }{4}\right)[/latex]

23. [latex]\left(0,\frac{\pi }{2}\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]

26. [latex]{r}^{2}=9\cos\theta [/latex]

27. [latex]r=\cos\left(\frac{\theta }{5}\right)[/latex]

28. [latex]r=2\sec\theta [/latex]

29. [latex]r=1+\cos\theta [/latex]

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

30. [latex]r=3[/latex]

31. [latex]\theta =\frac{\pi }{4}[/latex]

32. [latex]r=\sec\theta [/latex]

33. [latex]r=\csc\theta [/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]

36. [latex]x=8[/latex]

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

37. [latex]3x-y=2[/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]

40. [latex]r=6\cos\theta [/latex]

41. [latex]r=\theta [/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]

44. [latex]r=3 - 2\cos\theta [/latex]

45. [latex]r=2 - 2\sin\theta [/latex]

46. [latex]r=5 - 4\sin\theta [/latex]

47. [latex]r=3\cos\left(2\theta \right)[/latex]

48. [latex]r=3\sin\left(2\theta \right)[/latex]

49. [latex]r=2\cos\left(3\theta \right)[/latex]

50. [latex]r=3\cos\left(\frac{\theta }{2}\right)[/latex]

51. [latex]{r}^{2}=4\cos\left(2\theta \right)[/latex]

52. [latex]{r}^{2}=4\sin\theta [/latex]

53. [latex]r=2\theta [/latex]

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].

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].