If $Z_1$ and $Z_2$ are complex numbers such that $|Z_1+Z_2|=|Z_1|+|Z_2|$,then the difference in the amplitudes of $Z_1$ and $Z_2$ is

If $|z| = 4$ and $\text{arg}(z) = \frac{5\pi}{6}$,then $z =$

The amplitude of $0$ is

The amplitude of $\frac{1 + \sqrt{3}i}{\sqrt{3} + i}$ is

The amplitude of $\sin \frac{\pi}{5} + i(1 - \cos \frac{\pi}{5})$ is

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$. If $\operatorname{Arg}(x)=2 \alpha$,$\operatorname{Arg}(y)=3 \beta$,and $\alpha+\beta=\frac{\pi}{36}$,then $x^6 y^4+\frac{1}{x^6 y^4}=$