Question

Find the modulus and argument of the complex numbers $-1-i\sqrt{3}$

Answer 1

Modulus and Argument of $z=-1-i\sqrt{3}$

Step 1 : Given $z=-1-i\sqrt{3}$

Modulus of complex number, $r=\sqrt{x^2+y^2}$
$\Rightarrow r=\sqrt{(-1{)}^2+(-\sqrt{3}{)}^2}\Rightarrow r=\sqrt{1+3}\Rightarrow r=2$
$\therefore \text{modulus}r=2$

Step 3 : $\because z$ lies in the third quadrant
$\therefore \text{Argument}=-(\pi -{\tan }^{-1}\left|\frac{y}{x}\right|)=-(\pi -{\tan }^{-1}\left|\frac{-\sqrt{3}}{-1}\right|)$

$\therefore \text{Argument}=-\frac{2\pi }{3}$

Modulus and Argument of complex number $-1-i\sqrt{3}\text{are}2\text{and}-\frac{2\pi }{3}$ respectively.