Example 4. Write the complex number in trigonometric form

$$
z=-1-i \sqrt{3}
$$
Solution. We have

$$
|z|=\sqrt{(-1)^{2}+(-\sqrt{3})^{2}}=2 ; \quad \operatorname{tg} \varphi=\frac{-\sqrt{3}}{-1}=\sqrt{3}, \quad \varphi=-\frac{2}{3} \pi
$$

Therefore,

$$
-1-i \sqrt{3}=2\left[\cos \left(-\frac{2}{3} \pi\right)+i \sin \left(-\frac{2}{3} \pi\right)\right]
$$