How to find the amplitude of a complex number $z=-1-\sqrt{3}i$

Question

Find the amplitude of a complex number $z=-1-\sqrt{3}i$. I got the modulus of $z$ is $2$. After solving that I am getting $\cos(-60^\circ)$ and $\sin(-60^\circ)$.

I don’t have any idea how to solve after that

Answer 1

Draw a picture!

Here, $\operatorname{Arg}(z)$ denotes de amplitude of the complex number $z$, or better said, denotes the argument of $z$. Then, we see that $$\operatorname{Arg}(z) = 180^\circ + \theta$$ and, with a little bit of trigonometry, $\theta$ satisfies that $$\tan \theta = \frac{-\sqrt 3}{-1} = \sqrt 3$$ then, $\theta = \arctan(\sqrt 3) = 60^\circ$ and then $$\operatorname{Arg}(z) = 180^\circ + 60^\circ = 240^\circ.$$