Question

Find the multiplicative inverse of $\frac{\sqrt{3}}{2}-\frac{1}{2}i$.

Answer 1

Let $Z_i$ be the multiplicative Inverse of $\frac{\sqrt{3}}{2}-\frac{1}{2}i$
so that,
$Z_i(\frac{\sqrt{3}}{2}-\frac{1}{2}i)=1\Rightarrow Z_i=\frac{1}{\frac{\sqrt{3}}{2}-\frac{1}{2}i}=\frac{\frac{\sqrt{3}}{2}+\frac{1}{2}i}{(\frac{\sqrt{3}}{2}-\frac{1}{2}i)(\frac{\sqrt{3}}{2}+\frac{1}{2}i)}=\frac{\frac{\sqrt{3}}{2}+\frac{1}{2}i}{(\frac{\sqrt{3}}{2}{)}^2-(-{\frac{1}{2}}^2)}=\frac{\frac{\sqrt{3}}{2}+\frac{1}{2}i}{1}=\frac{\sqrt{3}}{2}+\frac{1}{2}i$