Question

Find the amplitude and modulus of following

(i) $\frac{21}{5}-\frac{12}{5}i$ (ii) $3+4i$

Answer 1

$\left(i\right)z=\frac{21}{5}-\frac{12i}{5}\left|z\right|=\sqrt{\frac{441}{25}+\frac{144}{25}}=\sqrt{\frac{585}{25}}=\sqrt{\frac{117}{5}}$

Amp(z)=Arg(z)$=-{\tan }^{-1}\left|\frac{Im\left(z\right)}{Re\left(z\right)}\right|$ ($\therefore z$ is in $4th$ quadrant)

$=-{\tan }^{-1}\left(\frac{12}{21}\right)=-{\tan }^{-1}\left(\frac{4}{7}\right)$

$\left(ii\right)z=3+4i\left|z\right|=\sqrt{3^2+4^2}=\sqrt{9+16}=5;$

Amp(z)=Arg(z)$={\tan }^{-1}\left|\frac{Im\left(z\right)}{Re\left(z\right)}\right|$

$={\tan }^{-1}\left(\frac{4}{3}\right)$