<s>[INST] Complex number and polar coordinates True or false: the polar coordinates of $-1-i$ are $-\sqrt{2}\operatorname{cis}\frac{\pi}{4}$
In my opinion it's true:
$\tan\theta=\frac{-1}{-1}=1\Rightarrow \theta=\frac{5\pi}{4}$, $r=\sqrt{(-1)^2+(-1)^2}=\sqrt{2}$
Therefore we get: $z=\sqrt{2}\operatorname{cis}\frac{5\pi}{4}=\sqrt{2}(\cos\frac{5\pi}{4}+i\sin\frac{5\pi}{4})=\sqrt{2}(-\cos\frac{\pi}{4}-i\sin\frac{\pi}{4})=-\sqrt{2}\operatorname{cis}\frac{\pi}{4}$
that means it's true. But it was written in the algebra book that because of $-2$, it's not polar coordinate. Is the proof right?
 [/INST] No, it's false. The $r$ part must be positive and is computed by
$$
r=\sqrt{(-1-i)(-1+i)}=\sqrt{1+1}=\sqrt{2}
$$
This already answers the true/false question. If you want to find the $\theta$ part, you need to find $\theta$ such that
$$
\cos\theta=-\frac{1}{\sqrt{2}},\quad\sin\theta=-\frac{1}{\sqrt{2}}
$$
and this is clearly
$$
\theta=\pi+\frac{\pi}{4}=\frac{5\pi}{4}
$$
</s>