Question

Find the modulus and argument of $z=\frac{3+2i}{-2+i}$.

Answer 1

$z=\frac{3+2i}{-2+i}$

$z=\frac{3+2i}{-2+c}\times \frac{(i+2)}{(i+2)}=\frac{3i+4i-2+6}{-1-4}$

$z=\frac{4+7i}{-5}$

$z=-\frac{4}{5}+\left(\frac{-7i}{5}\right)$

$121=\sqrt{\frac{16}{25}+\frac{49}{25}}=\sqrt{\frac{65}{25}}=\frac{\sqrt{65}}{5}$

$Arg\left(2\right)=-\pi +{\tan }^{-1}\left(\frac{7/5}{4/5}\right)$

$=-\pi +{\tan }^{-1}\left(\frac{7}{4}\right)$

$arg\left(2\right)=2k\pi -\pi +{\tan }^{-1}\left(7/4\right)$

$arg\left(2\right)=2(k-1)\pi +{\tan }^{-1}\left(7/4\right)$