$\text{sin⁡}\frac{2\mathrm{\pi }}{7}+\sin \frac{4\mathrm{\pi }}{7}+\sin \frac{8\mathrm{\pi }}{7}=$

Let $\mathrm{s}=\sin \frac{2\mathrm{\pi }}{7}+\sin \frac{4\mathrm{\pi }}{7}+\sin \frac{8\mathrm{\pi }}{7}$
$\begin{array}{l}\Rightarrow \mathrm{c}=\cos \frac{2\mathrm{\pi }}{7}+\cos \frac{4\mathrm{\pi }}{7}+\cos \frac{8\mathrm{\pi }}{7}\\ \Rightarrow \mathrm{c}+\mathrm{is}=\mathrm{\alpha }+{\mathrm{\alpha }}^2+{\mathrm{\alpha }}^4\dots .\mathrm{eq}\left(1\right)\end{array}$
where $\mathrm{\alpha }=\cos \frac{2\mathrm{\pi }}{7}+\mathrm{isin}\frac{2\mathrm{\pi }}{7}$ is complex 7th root of unity.
$\mathrm{c}-\mathrm{is}=\stackrel{\mathrm{\_}}{\mathrm{\alpha }}+{\stackrel{\mathrm{\_}}{\mathrm{\alpha }}}^2+{\stackrel{\mathrm{\_}}{\mathrm{\alpha }}}^4={\mathrm{\alpha }}^6+{\mathrm{\alpha }}^5+{\mathrm{\alpha }}^3\mathrm{eq}\left(2\right)$
Adding equation (1) and (2) then
$\begin{array}{l}2\mathrm{c}=\mathrm{\alpha }+{\mathrm{\alpha }}^2+{\mathrm{\alpha }}^4+{\mathrm{\alpha }}^6+{\mathrm{\alpha }}^5+{\mathrm{\alpha }}^3\\ =\mathrm{\alpha }+{\mathrm{\alpha }}^2+{\mathrm{\alpha }}^3+{\mathrm{\alpha }}^4+{\mathrm{\alpha }}^5+{\mathrm{\alpha }}^6\Rightarrow \mathrm{c}=\frac{-1}{2}\end{array}$
Also multiplaying eq (1) and (2)  ${\mathrm{c}}^2+{\mathrm{s}}^2=2$
$\Rightarrow {\mathrm{s}}^2=2-{\left(\frac{1}{2}\right)}^2\Rightarrow \mathrm{s}=\frac{\sqrt{7}}{2}$