Question

Express the complex number $\frac{(1+\sqrt{3}i{)}^2}{\sqrt{3}-i}$ in the form of $a+ib$. Hence, find the modulus and argument of the complex number

Answer 1

$z=\frac{(1+\sqrt{3}i{)}^2}{(\sqrt{3}-i)}$
$=\frac{1+3i^2+2\sqrt{3}i}{(\sqrt{3}-i)}$
$=\frac{1-3+2\sqrt{3}i}{\sqrt{3}-i}$
$=\frac{-2+2\sqrt{3}i}{\sqrt{3}-i}\times \frac{\sqrt{3}+i}{\sqrt{3}+i}$
$=\frac{-2\sqrt{3}+6i-2i+2\sqrt{3}{i}^2}{3-i^2}$
$=\frac{-4\sqrt{3}+4i}{3+1}$
$=\frac{4(-\sqrt{3}+i)}{4}$
$\therefore z=-\sqrt{3}+i$
Here, $a=-\sqrt{3},b=1$
Modulus, $|z|=\sqrt{a^2+b^2}$
$=\sqrt{(-\sqrt{3}{)}^2+(1{)}^2}$
$=\sqrt{3+1}=\sqrt{4}$
$=2$
$Argument=-\pi +{\tan }^{-1}\left|\frac{b}{a}\right|$
$=-\pi +{\tan }^{-1}\left|\frac{1}{-\sqrt{3}}\right|$
$=-\pi +{\tan }^{-1}\frac{1}{\sqrt{3}}$
$=-\pi +\frac{\pi }{6}=\frac{-5\pi }{6}$