14. Find the value of $\sin \frac{\pi}{7} \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}$.
14. Let $\sin \frac{\pi}{7} \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}=t$, then $t^{2}=\sin ^{2} \frac{\pi}{7} \sin ^{2} \frac{2 \pi}{7} \sin ^{2} \frac{3 \pi}{7}=\frac{1}{8}\left(1-\cos \frac{2 \pi}{7}\right)(1-$
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\begin{array}{l}
\left.\cos \frac{4 \pi}{7}\right)\left(1-\cos \frac{6 \pi}{7}\right)=\frac{1}{8}\left[1-\left(\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}\right)+\left(\cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}+\cos \frac{4 \pi}{7} \cos \frac{6 \pi}{7}+\right.\right. \\
\left.\left.\cos \frac{6 \pi}{7} \cos \frac{2 \pi}{7}\right)-\cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} \cos \frac{6 \pi}{7}\right]=\frac{1}{8}\left[1-\cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} \cos \frac{6 \pi}{7}\right] . \text { And } \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} \\
\cos \frac{6 \pi}{7}=\frac{8 \sin \frac{2}{7} \pi \cos \frac{2}{7} \pi \cos \frac{4 \pi}{7} \cos \frac{6}{7} \pi}{8 \sin \frac{2}{7} \pi}=\frac{1}{8}, \text { so } t^{2}=\frac{7}{8^{2}} . \text { Also, } \sin \frac{\pi}{7} \cdot \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}=t>0, \\
\text { then } t=\frac{\sqrt{7}}{8} .
\end{array}
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