Question

Find the modulus and argument of the given complex number $-\sqrt{3}+i$

Answer 1

Modulus and argument. $z=-\sqrt{3}+i$

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

Step 2 : Modulus of complex number $r=\sqrt{x^2+y^2}$
$\Rightarrow r=\sqrt{(-\sqrt{3}{)}^2+(1{)}^2}$
$\Rightarrow r=\sqrt{3+1}\Rightarrow r=2$

$\therefore \text{modulus}r=2$

Step 3 : $\because z$ lies in the second quadrant.
$\therefore \text{Argument}=(\pi -{\tan }^{-1}\left|\frac{y}{x}\right|)=(\pi -{\tan }^{-1}\left|\frac{1}{-\sqrt{3}}\right|)=\pi -\frac{\pi }{6}=\frac{5\pi }{6}$

$\therefore \text{Argument}=\frac{5\pi }{6}$

Modulus and Argument of complex number $-\sqrt{3}+i$ are $2$ and $\frac{5\pi }{6}$ respectively.