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

$\text{Let }s=\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{8\pi }{7}$

$\Rightarrow c=\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{8\pi }{7}$

$\Rightarrow c+is=\alpha +{\alpha }^2+{\alpha }^4\dots . \text{eq (1)}$

where  $\alpha =\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$

is complex 7th root of unity.

$c-is=\overline{\alpha }+{\overline{\alpha }}^2+{\overline{\alpha }}^4={\alpha }^6+{\alpha }^5+{\alpha }^3\mathrm{eq}\left(2\right)$

Adding equation (1) and (2) then

$2c=\alpha +{\alpha }^2+{\alpha }^4+{\alpha }^6+{\alpha }^5+{\alpha }^3$

$=\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6\Rightarrow c=\frac{-1}{2}$

Also multiplaying eq (1) and (2)  $c^2+s^2=2$

$\Rightarrow s^2=2-{\left(\frac{1}{2}\right)}^2\Rightarrow s=\frac{\sqrt{7}}{2}$