10.5E: Polar Form of Complex Numbers (Exercises)
For the following exercises, find the absolute value of each complex number.
28. \(-2+6 \mathbf{i}\)
29. \(4-3 \mathbf{i}\)
Write the complex number in polar form.
30. \(5+9 \mathbf{i}\)
31. \(\frac{1}{2}-\frac{\sqrt{3}}{2} \mathbf{i}\)
For the following exercises, convert the complex number from polar to rectangular form.
32. \(z=5 \operatorname{cis}\left(\frac{5 \pi}{6}\right)\)
33. \(z=3 \operatorname{cis}\left(40^{\circ}\right)\)
For the following exercises, find the product \(z_{1} z_{2}\) in polar form.
34.
\(z_{1}=2 \operatorname{cis}\left(89^{\circ}\right)\)
\(z_{2}=5 \operatorname{cis}\left(23^{\circ}\right)\)
35.
\(z_{1}=10\) cis \(\left(\frac{\pi}{6}\right)\)
\(z_{2}=6 \operatorname{cis}\left(\frac{\pi}{3}\right)\)
For the following exercises, find the quotient \(\frac{z_{1}}{z_{2}}\) in polar form.
36. \(z_{1}=12 \operatorname{cis}\left(55^{\circ}\right)\)
\(z_{2}=3 \operatorname{cis}\left(18^{\circ}\right)\)
37. \(z_{1}=27 \operatorname{cis}\left(\frac{5 \pi}{3}\right)\)
\(z_{2}=9 \operatorname{cis}\left(\frac{\pi}{3}\right)\)
For the following exercises, find the powers of each complex number in polar form.
38. Find \(z^{4}\) when \(z=2\) cis \(\left(70^{\circ}\right)\)
39. Find \(z^{2}\) when \(z=5\) cis \(\left(\frac{3 \pi}{4}\right)\)
For the following exercises, evaluate each root.
40. Evaluate the cube root of \(z\) when \(z=64\) cis \(\left(210^{\circ}\right)\).
41. Evaluate the square root of \(z\) when \(z=25\) cis \(\left(\frac{3 \pi}{2}\right)\).
For the following exercises, plot the complex number in the complex plane.
42. \(6-2 \mathbf{i}\)
43. \(-1+3 \mathbf{i}\)