La forme trigonométrique de $z_1=\sqrt{3}+ i$ ?

$|z_1|=\sqrt{\sqrt{3}^2+1^2}=2$

$\begin{aligned} z_1&=\sqrt{3}+ i=2\left[\dfrac{\sqrt{3}}{2}+i\dfrac{1}{2}\right]\\ &=2\left[\cos(\dfrac{\pi}{6})+i\sin(\dfrac{\pi}{6})\right] \end{aligned}$

La forme trigonométrique de $z_2=1 - i$ ?

$|z_2|=\sqrt{1^2+(-1)^2}=\sqrt{2}$

$\begin{aligned} z_2&=1-i=\sqrt{2}(\dfrac{1}{\sqrt{2}}-i\dfrac{1}{\sqrt{2}})\\ &=\sqrt{2}\left[\cos(\dfrac{\pi}{4})-i\sin(\dfrac{\pi}{4})\right] \\&=\sqrt{2}\left[\cos(-\dfrac{\pi}{4})+i\sin(-\dfrac{\pi}{4})\right] \end{aligned}$

La forme trigonométrique de $z_3=-\sqrt{3}+i$ ?

$|z_3|=\sqrt{(-\sqrt{3})^2+1^2}=2$

$\begin{aligned} z_3&=-\sqrt{3}+i=2\left[-\dfrac{\sqrt{3}}{2}+i\dfrac{1}{2}\right]\\ &=2\left[-\cos(\dfrac{\pi}{6})+i\sin(\dfrac{\pi}{6})\right] \\&=2\left[\cos(\pi-\dfrac{\pi}{6})+i\sin(\pi-\dfrac{\pi}{6})\right] \\&=2\left[\cos(\dfrac{5\pi}{6})+i\sin(\dfrac{5\pi}{6})\right] \end{aligned}$

La forme trigonométrique de $z_4=-2\sqrt{3} - 2i$ ?

$|z_4|=\sqrt{(-2\sqrt{3})^2+(-2)^2}=4$

$\begin{aligned} z_4&=-2\sqrt{3} - 2i\\\\ &=4\left[-\dfrac{2\sqrt{3}}{4}-i\dfrac{2}{4}\right]\\ \\&=4\left[-\dfrac{\sqrt{3}}{2}-i\dfrac{1}{2}\right] \\ \\&=4\left[-\cos(\dfrac{\pi}{6})-i\sin(\dfrac{\pi}{6})\right] \\ \\&=4\left[\cos(\pi+\dfrac{\pi}{6})+i\sin(\pi+\dfrac{\pi}{6})\right]\\\\ &=4\left[\cos(\dfrac{7\pi}{6})+i\sin(\dfrac{7\pi}{6})\right] \end{aligned}$