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<math> (1 + j) = \sqrt{2}e^{j\frac{\pi}{4}} </math> | <math> (1 + j) = \sqrt{2}e^{j\frac{\pi}{4}} </math> | ||
| − | <math> (1 + j)^{5} = (\sqrt{2}e^{j\frac{\pi}{4}})^{5} </math> | + | <math> (1 + j)^{5} = (\sqrt{2}e^{j\frac{\pi}{4}})^{5} = 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}}</math> |
= 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}} | = 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}} | ||
Revision as of 00:18, 13 June 2008
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}} $
= 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}}
