Problem Set: Polar Coordinates

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

1. $\left(3,\frac{\pi }{6}\right)$

2. $\left(-2,\frac{5\pi }{3}\right)$

3. $\left(0,\frac{7\pi }{6}\right)$

4. $\left(-4,\frac{3\pi }{4}\right)$

5. $\left(1,\frac{\pi }{4}\right)$

6. $\left(2,\frac{5\pi }{6}\right)$

7. $\left(1,\frac{\pi }{2}\right)$

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.

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 $\left(0,2\pi \right]$. Round to three decimal places.

12. $\left(2,2\right)$

13. $\left(3,-4\right)$

14. $\left(8,15\right)$

15. $\left(-6,8\right)$

16. $\left(4,3\right)$

17. $\left(3,\text{-}\sqrt{3}\right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

18. $\left(2,\frac{5\pi }{4}\right)$

19. $\left(-2,\frac{\pi }{6}\right)$

20. $\left(5,\frac{\pi }{3}\right)$

21. $\left(1,\frac{7\pi }{6}\right)$

22. $\left(-3,\frac{3\pi }{4}\right)$

23. $\left(0,\frac{\pi }{2}\right)$

24. $\left(-4.5,6.5\right)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$ -axis, the $y$ -axis, or the origin.

25. $r=3\sin\left(2\theta \right)$

26. ${r}^{2}=9\cos\theta$

27. $r=\cos\left(\frac{\theta }{5}\right)$

28. $r=2\sec\theta$

29. $r=1+\cos\theta$

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

30. $r=3$

31. $\theta =\frac{\pi }{4}$

32. $r=\sec\theta$

33. $r=\csc\theta$

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

34. ${x}^{2}+{y}^{2}=16$

35. ${x}^{2}-{y}^{2}=16$

36. $x=8$

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

37. $3x-y=2$

38. ${y}^{2}=4x$

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

39. $r=4\sin\theta$

40. $r=6\cos\theta$

41. $r=\theta$

42. $r=\cot\theta \csc\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

43. $r=1+\sin\theta$

44. $r=3 - 2\cos\theta$

45. $r=2 - 2\sin\theta$

46. $r=5 - 4\sin\theta$

47. $r=3\cos\left(2\theta \right)$

48. $r=3\sin\left(2\theta \right)$

49. $r=2\cos\left(3\theta \right)$

50. $r=3\cos\left(\frac{\theta }{2}\right)$

51. ${r}^{2}=4\cos\left(2\theta \right)$

52. ${r}^{2}=4\sin\theta$

53. $r=2\theta$

54. [T] The graph of $r=2\cos\left(2\theta \right)\sec\left(\theta \right)$. 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 $r=\frac{6}{2\sin\theta -3\cos\theta }$.

56. [T] Use a graphing utility to graph $r=\frac{1}{1-\cos\theta }$.

57. [T] Use technology to graph $r={e}^{\sin\left(\theta \right)}-2\cos\left(4\theta \right)$.

58. [T] Use technology to plot $r=\sin\left(\frac{3\theta }{7}\right)$ (use the interval $0\le \theta \le 14\pi$).

59. Without using technology, sketch the polar curve $\theta =\frac{2\pi }{3}$.

60. [T] Use a graphing utility to plot $r=\theta \sin\theta$ for $\text{-}\pi \le \theta \le \pi$.

61. [T] Use technology to plot $r={e}^{-0.1\theta }$ for $-10\le \theta \le 10$.

62. [T] There is a curve known as the “Black Hole.” Use technology to plot $r={e}^{-0.01\theta }$ for $-100\le \theta \le 100$.

63. [T] Use the results of the preceding two problems to explore the graphs of $r={e}^{-0.001\theta }$ and $r={e}^{-0.0001\theta }$ for $|\theta |>100$.