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Example Express the following in the polar form．

\begin{displaymath}\begin{array}{ll} 1. & z = 3 - \sqrt{3}i \\ 2. & z = 2i \end{array}\end{displaymath}

Answer When we express complex number $a+bi$ using the distance $r$ and the argument $\theta$, it is called polar form, where $x = r\cos{\theta}, y = r\sin{\theta}$. Then we have $x+iy = r(\cos{\theta} + i \sin{\theta})$.

1. To express $z = 3 - \sqrt{3}i$ in polar form, we have $r = \sqrt{3^2 + (\sqrt{3})^2} = \sqrt{12} = 2\sqrt{3}$. Also, $\theta = \tan^{-1}{\frac{-\sqrt{3}}{3}} = -\frac{\pi}{6}$. Then，

$\displaystyle 2\sqrt{3}(\cos(\frac{-\pi}{6}) + i \sin(\frac{-\pi}{6}). $

2. To express $z =2i$ in polar form，we have $r = \sqrt{0 +2^2} = \sqrt{4} = 2$. Also, $\theta = \tan^{-1}{\infty} = \frac{\pi}{2}$. Thus，

$\displaystyle 2(\cos{\frac{\pi}{2}} + i \sin{\frac{\pi}{2}}).$