The argument of $\frac{1+i \sqrt{3}}{\sqrt{3}+i}$,where $i=\sqrt{-1}$,is

If the roots of the equation $z^2-i=0$ are $\alpha$ and $\beta$,then $|\operatorname{Arg} \beta-\operatorname{Arg} \alpha|=$

If $\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in [0, 2 \pi],$ is a real number,then an argument of $\sin \theta + i \cos \theta$ is

If $z = 1 - \cos \alpha + i \sin \alpha $,then $\text{amp } z$ =

If $z=x+iy$,where $x, y \in \mathbb{R}$ and the point $P$ in the Argand plane represents $z$,then the locus of $P$ satisfying the condition $\arg \left(\frac{z-1}{z-3i}\right)=\frac{\pi}{2}$ is:

Let $z$ and $w$ be two complex numbers such that $\bar{z}+i \bar{w}=0$ and $\operatorname{Arg}(z w)=\pi$. Then,$\operatorname{Arg} z=$