Question

Find the value of the principal argument of the complex number $z=\frac{(1+i\sqrt{3}{)}^2}{(1-i{)}^3}$.

Answer 1

Given $z=\frac{(1+i\sqrt{3}{)}^2}{(1+i)3}=\frac{(1+i\sqrt{3})(1+i\sqrt{3})}{(1-i)(1-i)(1-i)}$

$\therefore arg\left(z\right)=\frac{{\tan }^{-1}\sqrt{3}+{\tan }^{-1}\sqrt{3}}{{\tan }^{-1}(-1)+{\tan }^{-1}(-1)+{\tan }^{-1}(-1)}$

$=2{\tan }^{-1}\sqrt{3}-3{\tan }^{-1}(-1)$

$=120+3{\tan }^{-1}\left(1\right)$ $(\because (1-i\left)lieinIVquad\right)$

$arg\left(z\right)={255}^o$ $\therefore \theta =-{\tan }^{-1}\left(1\right)$