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Example Express the following equation using Euler's formula.．

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

Answer When we express a complex number $a+bi$ by using the distance from the origin $r$ and the angle from the $x$-axis called argument $\theta$ in the following way: $x = r\cos{\theta}, y = r\sin{\theta}$．we call this representation polar form. So, $x+iy = r(\cos{\theta} + i \sin{\theta})$. $r(\cos{\theta} + i \sin{\theta})　= e^{i\theta}$ is called Euler's formula.．

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

$\displaystyle 2e^{-\frac{\pi}{6}i}. $

2. Express $z =2i$ in the polar form， $r = \sqrt{0 +2^2} = \sqrt{4} = 2$. Also， $\theta = \tan^{-1}{\infty} = \frac{\pi}{2}$. Thsu,