DeMoivre's theorem and Euler's formula

Theorem 1..1 Euler's formula

$\displaystyle e^{i\theta} = \cos{\theta} + i \sin{\theta}$

Proof The MacLaurin expansion of $e^{x}, \sin{x}, \cos{x}$ are

$\displaystyle e^{x}$	$\displaystyle =$	$\displaystyle 1 + x + \frac{x^{2}}{2!} + \cdots = \sum_{n=0}^{\infty}\frac{x^{n}}{n!}$
$\displaystyle \sin{x}$	$\displaystyle =$	$\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}$
$\displaystyle \cos{x}$	$\displaystyle =$	$\displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$

For any real number $\theta$ , we have

$\displaystyle e^{i\theta}$	$\displaystyle =$	$\displaystyle (1 + i\theta + \frac{i^{2}\theta^2}{2!} + \cdots$
	$\displaystyle =$	$\displaystyle 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots) + i(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots)$
	$\displaystyle =$	$\displaystyle \cos{\theta} + i\sin{\theta}$

Properties

(1) $\overline{z_{1} \pm z_{2}} = \overline{z_{1}} \pm \overline{z_{2}}, \overline... ...}}, \overline{\frac{z_{1}}{z_{2}}} = \frac{\overline{z_{1}}}{\overline{z_[2}}$

(2) $\Re z = \frac{z + \overline{z}}{2}, \Im z = \frac{(z - \overline{z})}{2i}$

(3) $\vert z_{1}z_{2}\vert = \vert z_{1}\vert\vert z_{2}\vert, \vert\frac{z_{1}}{z_{2}}\vert = \frac{\vert z_{1}\vert}{\vert z_{2}\vert} (z_{2} \neq 0)$

(4) $\arg(z_{1}z_{2}) = \arg z_{1} + \arg z_{2}, \arg(\frac{z_{1}}{z_{2}}) = \arg z_{1} - \arg z_{2}$

(5) $\vert z\vert = \vert\overline{z}\vert, z\overline{z} = \vert z\vert^{2}, \arg{z} = -\arg{z}$

(6) $\vert\vert z_{1}\vert - \vert z_{2}\vert\vert \leq \vert z_{1} \pm z_{2}\vert \leq \vert z_{1}\vert + \vert z_{2}\vert$

(7) $z = 0$ and $\vert z\vert = 0$ are equivalent ( $\arg 0$ is undetermined)

(8) $(\cos{\theta} + i\sin{\theta})^{n} = \cos{n\theta} + i\sin{n\theta} (n$ is constant $)$ De Moivre's theorem (9) Binary equation $z^{n} = \alpha(= re^{i\theta})$ has the following solutions