Question

Express in the form of $a+ib,a,b,\in R$ and $i=\sqrt{-1}$ then find conjugate modulus and amplitude of the complex number
$\frac{1+3i}{2-i}+\frac{1+2i}{2+i}$

Answer 1

Given,

$\frac{1+3i}{2-i}+\frac{1+2i}{2+i}$

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

$=\frac{3+10i}{5}$

$\therefore z=\frac{3}{5}+i2$

$|z|=\sqrt{{\left(\frac{3}{5}\right)}^2+2^2}$

$=\frac{\sqrt{109}}{5}$

$amp=\theta ={\tan }^{-1}\left(\frac{2}{\frac{3}{5}}\right)\Rightarrow \theta ={73.30}^{\circ }$