Question

Express $\frac{2+i}{(1+i)(1-2i)}$ in the form of $a+ib$. Find its modulus and argument.

Answer 1

Here $\frac{2+i}{(1+i)(1-2i)}=\frac{2+i}{1+i-2i-2i^2}=\frac{2+i}{3-i}$

$=\frac{2+i}{3-i}\times \frac{3+i}{3+i}=\frac{6+3i+2i+i^2}{9+1}$

$=\frac{5+5i}{10}=\frac{1}{2}+\frac{1}{2}i=(\frac{1}{2},\frac{1}{2})$

$\therefore$ Modulus $=\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2}=\frac{1}{\sqrt{2}}$
Since $(\frac{1}{2},\frac{1}{2})$ lies in the first quadrant
$\therefore$ Argument ($\alpha$)$={\tan }^{-1}\left|\frac{1/2}{1/2}\right|={\tan }^{-1}(-1)=\frac{\pi }{4}$