Question

Represent the complex number $-1-i$ in the polar form

Answer 1

$z=-1-i$

$\left|z\right|=\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2}=\sqrt{2}$

$\tan \varphi =\left|\frac{y}{x}\right|=\left|\frac{-1}{-1}\right|=\left|\frac{-1}{-1}\right|=1=\tan {45}^{\circ }$

$=\frac{\pi }{4}$

So, argument $=\frac{\pi }{4}-\pi =\frac{-3\pi }{4}$

$z=r\left(\cos \theta +i\sin \theta \right)$

$=\sqrt{2}\left[\cos \left(\frac{-3\pi }{4}\right)+\sin \left(\frac{-3\pi }{4}\right)\right]$