Solving roots of complex number

Given $z= \dfrac{i-1}{2}$. Evaluate $z^{1/2}$ and show the roots.

I got $z^{1/2}=\dfrac{1}{2^{1/4}}\left(\cos\dfrac{3\pi}{8} + i\sin\dfrac{3\pi}{8}\right)$

First of all is my $z^{1/2}$ correct? And secondly how to show the roots??

I would be glad and appreciated if someone could help me out.

Answer

$z=-\frac12+\frac i 2=R(\cos y+i\sin y)(say)$ where $R>0$

$R\cos y=-\frac12, R\sin y=\frac12\implies R^2=(-\frac12)^2+(\frac12)^2=\frac12$

So, $\tan y=\frac{R\sin y}{R\cos y}=-1\implies y=2n\pi+\frac{3\pi}{4}$ as $\cos y<0$ and $\sin y>0$ ,so $y$ lies in the 2nd quadrant.

So, $z =\frac 1{\sqrt2}(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4})$

So, the general value of $z^{\frac12}$ is $(\frac12)^{\frac14}(\cos(n\pi+\frac{3\pi}{8})+i\sin(n\pi+\frac{3\pi}{8}))$ where $n=0,1$
using de Moivre's formula.

$n=0\implies z^{\frac12}=(\frac12)^{\frac14}(\cos(\frac{3\pi}{8})+i\sin(\frac{3\pi}{8}))$

$n=1\implies z^{\frac12}=(\frac12)^{\frac14}(\cos(\pi+\frac{3\pi}{8})+i\sin(\pi+\frac{3\pi}{8}))=-(\frac12)^{\frac14}(\cos(\frac{3\pi}{8})+i\sin(\frac{3\pi}{8}))$

One may look into this, for the values of $n$.