Solutions 59: Polar Coordinates

Solutions to Odd-Numbered Exercises

1. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

3. Determine [latex]\theta[/latex] for the point, then move [latex]r[/latex] units from the pole to plot the point. If [latex]r[/latex] is negative, move [latex]r[/latex] units from the pole in the opposite direction but along the same angle. The point is a distance of [latex]r[/latex] away from the origin at an angle of [latex]\theta[/latex] from the polar axis.

5. The point [latex]\left(-3,\frac{\pi }{2}\right)[/latex] has a positive angle but a negative radius and is plotted by moving to an angle of [latex]\frac{\pi }{2}[/latex] and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point [latex]\left(3,-\frac{\pi }{2}\right)[/latex] has a negative angle and a positive radius and is plotted by first moving to an angle of [latex]-\frac{\pi }{2}[/latex] and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

7. [latex]\left(-5,0\right)[/latex]

9. [latex]\left(-\frac{3\sqrt{3}}{2},-\frac{3}{2}\right)[/latex]

11. [latex]\left(2\sqrt{5}, 0.464\right)[/latex]

13. [latex]\left(\sqrt{34},5.253\right)[/latex]

15. [latex]\left(8\sqrt{2},\frac{\pi }{4}\right)[/latex]

17. [latex]r=4\csc \theta[/latex]

19. [latex]r=\sqrt[3]{\frac{sin\theta }{2co{s}^{4}\theta }}[/latex]

21. [latex]r=3\cos \theta[/latex]

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

25. [latex]r=\frac{9\sin \theta }{{\cos }^{2}\theta }[/latex]

27. [latex]r=\sqrt{\frac{1}{9\cos \theta \sin \theta }}[/latex]

29. [latex]{x}^{2}+{y}^{2}=4x[/latex] or [latex]\frac{{\left(x - 2\right)}^{2}}{4}+\frac{{y}^{2}}{4}=1[/latex]; circle

31. [latex]3y+x=6[/latex]; line

33. [latex]y=3[/latex]; line

35. [latex]xy=4[/latex]; hyperbola

37. [latex]{x}^{2}+{y}^{2}=4[/latex]; circle

39. [latex]x - 5y=3[/latex]; line

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

43. [latex]\left(5,\pi \right)[/latex]

45.
Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi/2 (closer to 3pi/2).

47.
Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi/2 and 2pi.

49.
Polar coordinate system with a point located on the fifth concentric circle and pi/2.

51.
Polar coordinate system with a point located on the third concentric circle and 2/3 of the way between pi/2 and pi (closer to pi).

53.
Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi/2.

55. [latex]r=\frac{6}{5\cos \theta -\sin \theta }[/latex]
Plot of given line in the polar coordinate grid

57. [latex]r=2\sin \theta[/latex]
Plot of given circle in the polar coordinate grid

59. [latex]r=\frac{2}{\cos \theta }[/latex]
Plot of given circle in the polar coordinate grid

61. [latex]r=3\cos \theta[/latex]
Plot of given circle in the polar coordinate grid.

63. [latex]{x}^{2}+{y}^{2}=16[/latex]
Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.

65. [latex]y=x[/latex]
Plot of line y=x in the rectangular coordinates grid.

67. [latex]{x}^{2}+{\left(y+5\right)}^{2}=25[/latex]
Plot of circle with radius 5 centered at (0,-5).

69. [latex]\left(1.618,-1.176\right)[/latex]

71. [latex]\left(10.630,131.186^\circ \right)[/latex]

73. [latex]\left(2,3.14\right)or\left(2,\pi \right)[/latex]

75. A vertical line with [latex]a[/latex] units left of the y-axis.

77. A horizontal line with [latex]a[/latex] units below the x-axis.

79.
Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.

81.
Graph of ray starting at (2, pi/4) and extending in a positive direction along pi/4 - polar coordinate grid.

83.
Graph of the shaded region 0 to pi/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid