Polar Coordinates

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]

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.

49.

51.

55. [latex]r=\frac{6}{5\cos \theta -\sin \theta }[/latex]

57. [latex]r=2\sin \theta[/latex]

59. [latex]r=\frac{2}{\cos \theta }[/latex]

61. [latex]r=3\cos \theta[/latex]

63. [latex]{x}^{2}+{y}^{2}=16[/latex]

65. [latex]y=x[/latex]

67. [latex]{x}^{2}+{\left(y+5\right)}^{2}=25[/latex]

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]

Polar Coordinates: Graphs

1. Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] is similar to symmetry about the [latex]y[/latex] -axis.

3. Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at [latex]\theta =0,\frac{\pi }{2},\pi \text{and }\frac{3\pi }{2}[/latex], and sketch the graph.

5. The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

7. symmetric with respect to the polar axis

9. symmetric with respect to the polar axis, symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex], symmetric with respect to the pole

11. no symmetry

13. no symmetry

15. symmetric with respect to the pole

17. circle

19. cardioid

21. cardioid

23. one-loop/dimpled limaçon

25. one-loop/dimpled limaçon

27. inner loop/two-loop limaçon

29. inner loop/two-loop limaçon

31. inner loop/two-loop limaçon

33. lemniscate

35. lemniscate

37. rose curve

39. rose curve

41. Archimedes’ spiral

43. Archimedes’ spiral

45.

47.

49.

51.

53.

61. The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

63. The graphs are spirals. The smaller the coefficient, the tighter the spiral.

65. [latex]\left(4,\frac{\pi }{3}\right),\left(4,\frac{5\pi }{3}\right)[/latex]

67. [latex]\left(\frac{3}{2},\frac{\pi }{3}\right),\left(\frac{3}{2},\frac{5\pi }{3}\right)[/latex]

69. [latex]\left(0,\frac{\pi }{2}\right),\left(0,\pi \right),\left(0,\frac{3\pi }{2}\right),\left(0,2\pi \right)[/latex]

71. [latex]\left(\frac{\sqrt[4]{8}}{2},\frac{\pi }{4}\right),\left(\frac{\sqrt[4]{8}}{2},\frac{5\pi }{4}\right)[/latex]
and at [latex]\theta =\frac{3\pi }{4},\frac{7\pi }{4}[/latex] since [latex]r[/latex] is squared

Polar Form of Complex Numbers

1. a is the real part, b is the imaginary part, and [latex]i=\sqrt{−1}[/latex]

3. Polar form converts the real and imaginary part of the complex number in polar form using [latex]x=r\cos\theta[/latex] and [latex]y=r\sin\theta[/latex]

7. [latex]5\sqrt{2}[/latex]

9. [latex]\sqrt{38}[/latex]

11. [latex]\sqrt{14.45}[/latex]

13. [latex]4\sqrt{5}\text{cis}\left(333.4^{\circ}\right)[/latex]

15. [latex]2\text{cis}\left(\frac{\pi}{6}\right)[/latex]

17. [latex]\frac{7\sqrt{3}}{2}+i\frac{7}{2}[/latex]

19. [latex]−2\sqrt{3}−2i[/latex]

21. [latex]−1.5−i\frac{3\sqrt{3}}{2}[/latex]

23. [latex]4\sqrt{3}\text{cis}\left(198^{\circ}\right)[/latex]

25. [latex]\frac{3}{4}\text{cis}\left(180^{\circ}\right)[/latex]

27. [latex]5\sqrt{3}\text{cis}\left(\frac{17\pi}{24}\right)[/latex]

29. [latex]7\text{cis}\left(70^{\circ}\right)[/latex]

31. [latex]5\text{cis}\left(80^{\circ}\right)[/latex]

33. [latex]5\text{cis}\left(\frac{\pi}{3}\right)[/latex]

35. [latex]125\text{cis}\left(135^{\circ}\right)[/latex]

37. [latex]9\text{cis}\left(240^{\circ}\right)[/latex]

39. [latex]\text{cis}\left(\frac{3\pi}{4}\right)[/latex]

41. [latex]3\text{cis}\left(80^{\circ}\right)\text{, }3\text{cis}\left(200^{\circ}\right)\text{, }3\text{cis}\left(320^{\circ}\right)[/latex]

43. [latex]2\sqrt[3]{4}\text{cis}\left(\frac{2\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{8\pi}{9}\right)\text{, }2\sqrt[3]{4}\text{cis}\left(\frac{14\pi}{9}\right)[/latex]

45. [latex]2\sqrt{2}\text{cis}\left(\frac{7\pi}{8}\right)\text{, }2\sqrt{2}\text{cis}\left(\frac{15\pi}{8}\right)[/latex]

47.

49.

51.

53.

55.

57. [latex]3.61e^{−0.59i}[/latex]

59. [latex]−2+3.46i[/latex]

61. [latex]−4.33−2.50i[/latex]