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.

2. 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.

3. 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.

4. [latex]\left(-5,0\right)[/latex]

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

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

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

8. [latex]r=4\csc \theta[/latex]

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

10. [latex]r=3\cos \theta[/latex]

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

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

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

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

15. [latex]3y+x=6[/latex]; line

16. [latex]y=3[/latex]; line

17. [latex]xy=4[/latex]; hyperbola

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

19. [latex]x - 5y=3[/latex]; line

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

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

22.

23.

24.

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

26. [latex]r=2\sin \theta[/latex]

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

28. [latex]r=3\cos \theta[/latex]

29. [latex]{x}^{2}+{y}^{2}=16[/latex]

30. [latex]y=x[/latex]

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

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

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

34. [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.

2. 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.

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

4. symmetric with respect to the polar axis

5. 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

6. no symmetry

7. no symmetry

8. symmetric with respect to the pole

9. circle

10. cardioid

11. cardioid

12. one-loop/dimpled limaçon

13. one-loop/dimpled limaçon

14. inner loop/two-loop limaçon

15. inner loop/two-loop limaçon

16. inner loop/two-loop limaçon

17. lemniscate

18. lemniscate

19. rose curve

20. rose curve

21. Archimedes’ spiral

22. Archimedes’ spiral

23.

24.

25.

26.

27.

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

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

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

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

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

33. [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]

2. 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]

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

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

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

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

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

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

9. [latex]−2\sqrt{3}−2i[/latex]

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

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

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

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

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

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

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

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

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

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

20. [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]

21. [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]

22. [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]

23.

24.

25.

26.

27.

28. [latex]3.61e^{−0.59i}[/latex]

29. [latex]−2+3.46i[/latex]

30. [latex]−4.33−2.50i[/latex]