Question

Find the modulus and the arguments of each of the complex numbers
$z=-\sqrt{3}+i$

Answer 1

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

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

Squaring both sides of (i) and adding

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

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

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

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

Since $sin\theta$ is positive and $cos\theta$ is negative

$\therefore \theta$ lies second quadrant

$\therefore \theta =(\pi -\frac{\pi }{6})=\frac{5\pi }{6}$

$\therefore |z|=2$ and arg(z)$=\frac{5\pi }{6}$