8.2.1: Polar Coordinates (Exercises)
section 8.2 exercises
Convert the given polar coordinates to Cartesian coordinates.
1. \((7, \dfrac{7\pi}{6})\)
2. \(6, \dfrac{3pi}{4}\)
3. \(4, \dfrac{7\pi}{4}\)
4. \(9, \dfrac{4\pi}{3}\)
5. \(6, -\dfrac{\pi}{4}\)
6. \(12, -\dfrac{\pi}{3}\)
7. \(3, \dfrac{\pi}{2}\)
8. \(5, \pi\)
9.\(-3, \dfrac{\pi}{6}\)
10. \(-2, \dfrac{2\pi}{3}\)
11. (3, 2)
12. (7, 1)
Convert the given Cartesian coordinates to polar coordinates.
13. (4, 2)
14. (8, 8)
15. (-4, 6)
16. (-5, 1)
17. (3, -5)
18. (6, -5)
19. (-10, -13)
20. (-4, -7)
Convert the given Cartesian equation to a polar equation.
21. \(x = 3\)
22. \(y = 4\)
23. \(y = 4x^2\)
24. \(y = 2x^4\)
25. \(x^2 = y^2 = 4y\)
26. \(x^2 = y^2 = 3x\)
27. \(x^2 - y^2 = x\)
28. \(x^2 - y^2 = 3y\)
Convert the given polar equation to a Cartesian equation.
29. \(r = 3\sin(\theta)\)
30. \(r = 4\cos(\theta)\)
31. \(r = \dfrac{4}{\sin(\theta) + 7\cos(\theta)}\)
32. \(r = \dfrac{6}{\cos(\theta) + 3\sin(\theta)}\)
33. \(r = 2\sec(\theta)\)
34. \(r = 3\csc(\theta)\)
35. \(r = \sqrt{r\cos(\theta) + 2}\)
36. \(r^2 = 4 \sec(\theta)\csc(\theta)\)
Match each equation with one of the graphs shown.
37. \(r = 2 + 2\cos(\theta)\)
38. \(r = 2 + 2\sin(\theta)\)
39. \(r = 4 + 3\cos(\theta)\)
40. \(r = 3 + 4\cos(\theta)\)
41. \(r = 5\)
42. \(r = 2\sin(\theta)\)
Match each equation with one of the graphs shown.
43. \(r = \text{log}(\theta)\)
44. \(r = \theta \cos(\theta)\)
45. \(r = \cos(\dfrac{\theta}{2})\)
46. \(r = \sin(\theta)\cos^2(\theta)\)
47. \(r = 1 + 2\sin(3\theta)\)
48. \(r = 1 + \sin(2\theta)\)
Sketch a graph of the polar equation.
49. \(r = 3\cos(\theta)\)
50. \(r = 4\sin(\theta)\)
51. \(r = 3\sin(2\theta)\)
52. \(r = 4\sin(4\theta)\)
53. \(r = 5\sin(3\theta)\)
54. \(r = 4\sin(5\theta)\)
55. \(r = 3\cos(2\theta)\)
56. \(r = 4\cos(4\theta)\)
57. \(r = 2+ 2\cos(\theta)\)
58. \(r = 3 + 3\sin(\theta)\)
59. \(r = 1 + 3\sin(\theta)\)
60. \(r = 2 + 4 \cos(\theta)\)
61. \(r = 2\theta\)
62. \(r = \dfrac{1}{\theta}\)
63. \(r = 3 + \sec(\theta)\), a conchoid
64. \(r = \dfrac{1}{\sqrt{\theta}}\), a lituus
65. \(r = 2\sin(\theta)\tan(\theta)\), a cissoid
66. \(r = 2\sqrt{1- \sin^2(\theta)}\). a hippopede
- Answer
-
1. \((-\dfrac{7\sqrt{3}}{2}, -\dfrac{7}{2})\)
3. \((2\sqrt{2}, -2\sqrt{2})\)
5. \((3\sqrt{2}, -3\sqrt{2})\)
7. (0, 3)
9. \((-\dfrac{3\sqrt{3}}{2}, -\dfrac{3}{2}\)
11. (−1.248, 2.728)
13. \((2\sqrt{5}, 0.464)\)
15. \((2\sqrt{13}, 2.159)\)
17. \((\sqrt{34}, 5.253)\)
19. \((\sqrt{269}, 4.057)\)
21. \(r = 3\sec(\theta)\)
23. \(r = \dfrac{\sin(\theta)}{4\cos^(2)(\theta)}\)
25. \(r = 4\sin(\theta)\)
27. \(r = \dfrac{\cos(\theta)}{(\cos^(2)(\theta) - \sin^(2)(\theta))}\)
29. \(x^2 + y^2 = 3y\)
31. \(y = 7x = 4\)
33. \(x = 2\)
35. \(x^2 + y^2 = x + 2\)
37. A
39. C
41. E
43. C
45. D
47. F
49.
51.
53.
55.
57.
59.
61.
63.
65.