Express each of the following complex numbers in polar form, and plot them in the complex plane, indicating the magnitude and angle of each number.
A) $ 1 + j\sqrt{3} $
$ r = \sqrt{1^2 + \sqrt{3}^2} = \sqrt{4} = 2 $
$ \theta = arctan(\sqrt{3}/1) = arctan(\sqrt{3}) = \frac{\pi}{3} $
Therefore the polar form of this complex number is: $ 2e^{j\frac{\pi}{3}} $
B) $ -5 $
$ r = 5 $
$ \theta = \pi $
C) $ (1 + j)^{5} $
$ r = \sqrt{1^2 + 1^2} = \sqrt{2} $
$ \theta = \frac{\pi}{4} $
$ (1 + j) = \sqrt{2}e^{j\frac{\pi}{4}} $
$ (1 + j)^{5} = (\sqrt{2}e^{j\frac{\pi}{4}})^{5} = 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}} = 4\sqrt{2}e^{j(\pi + \frac{\pi}{4})} =4\sqrt{2}e^{j\pi}e^{j\frac{\pi}{4}} = 4(\sqrt{2}e^{j\frac{\pi}{4}}) = -4(1 + j) $
$ = 4\sqrt{2}e^{j(\pi + \frac{\pi}{4})} =4\sqrt{2}e^{j\pi}e^{j\frac{\pi}{4}} $
$ = 4(\sqrt{2}e^{j\frac{\pi}{4}}) = -4(1 + j) $
