If z is a non-zero complex numbers then the minimum value of $\frac{\mathrm{Im}\left(z^5\right)}{{\left(\mathrm{Im}z\right)}^5}$

$$\begin{gathered} z is a non-real complex number. \\ let z=r{e}^{ i\theta } \end{gathered}$$
$\frac{\mathrm{Im}{z}^5}{(\mathrm{Im}z{)}^5}=\frac{r^5\sin 5\theta }{r^5{\sin }^5\theta }$
$f\left(\theta \right)=\frac{\sin 5\theta }{{\sin }^5\theta }$
$\begin{array}{l}{\mathrm{f}}^{\mathrm{\prime }}\left(\theta \right)=0\\ \Rightarrow {\sin }^5\theta \cdot 5\cos 5\theta -\sin 5\theta \cdot 5{\sin }^4\theta \cdot \cos \theta \\ \Rightarrow \sin \theta \cdot \cos 5\theta =\cos \theta \cdot \sin 5\theta \\ \Rightarrow \tan \theta =\tan 5\theta \\ \Rightarrow \theta =0,\pi /4\dots \end{array}$
$f(\pi /4)=-4$
$\therefore \text{ Minimum value of }\frac{\mathrm{Imz}{}^5}{(\mathrm{Imz}{)}^5}\text{ is }-4$