Question

Find the modulus and the arguments of each of the complex numbers

$z=-1-i\sqrt{3}$

Answer 1

Here $z=-1-i\sqrt{3}=r(cos\theta +isin\theta )$

$\Rightarrow rcos\theta =-1$ and $rsin\theta =-\sqrt{3}\dots \left(i\right)$

Squaring both sides of (i) and adding

$r^2(co{s}^2\theta +si{n}^2\theta )=1+3$

$\Rightarrow r^2=4\Rightarrow r=2$

$\therefore 2cos\theta =-1$ and $2sin\theta =-\sqrt{3}$

$\Rightarrow cos\theta =\frac{-1}{2}$ and $sin\theta =\frac{-\sqrt{3}}{2}$

Since both $sin\theta$ and $cos\theta$ are negatie.

$\therefore \theta$ lies in third quadrant.

$\therefore \theta =(-\pi +\frac{\pi }{3})=\frac{-2\pi }{3}$

$\therefore |z|=2andarg\left(z\right)=\frac{-2\pi }{3}$