Find the argument of k

Question

Two complex numbers, $\tilde{k}$ and $̃$ are related as follows: $=̃/$ where, $ = \sqrt{−1}$ and $$ is a scalar. Given principal argument of $̃$, $Arg(̃) =-2/3$, , then find the principal argument of $\tilde{}$ (rounded-off to two decimal places).

Answer 1

The calculation $$\operatorname{Arg}\tilde{k}=\operatorname{Arg}\tilde{\varepsilon}-\operatorname{Arg}(i\omega)=\left\{ \begin{array}{cc} -\frac{2\pi}{3}-\frac{\pi}{2}=-\frac{7\pi}{6} & \omega>0\\ -\frac{2\pi}{3}+\frac{\pi}{2}=-\frac{\pi}{6} & \omega<0 \end{array}\right.$$ is almost right, but for principal arguments the $\omega>0$ result is modified to $-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}$.