Sine Summation

Let $\alpha=e^{2\pi i/7}$ .

Then $\sin\frac{2\pi}{7}=\frac{\alpha-\alpha^{-1}}{2i}\ ,\quad \sin\frac{4\pi}{7}=\frac{\alpha^2-\alpha^{-2}}{2i}\ ,\quad \sin\frac{8\pi}{7}=\frac{\alpha^4-\alpha^{-4}}{2i}\ .$

Also note that: $\alpha^7=1\quad\hbox{and}\quad \alpha^6+\alpha^5+\cdots+\alpha=-1$

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

$S = \frac{\alpha+\alpha^2+\alpha^4-(\alpha^3+\alpha^5+\alpha^6)}{2i}$

$S^2 = \frac{(\alpha+\alpha^2+\alpha^4)^2+(\alpha^3+\alpha^5+\alpha^6)^2 - 2(\alpha+\alpha^2+\alpha^4)(\alpha^3+\alpha^5+\alpha^6)}{-4}$

This simplifies to

$S^2=\frac{7}{4}$

Also, it can be easily seen that $S>0$ , hence we consider only the positive root:

$S=\frac{\sqrt7}{2}$

Hence, $a=7, b=2$ and $a+b=9$ .