Write the value of \( \sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10} \)
Solution:
Using identity,
\[
\sin x\sin(60^\circ-x)\sin(60^\circ+x)
=
\frac14\sin3x
\]
Here,
\[
x=\frac{\pi}{15}=12^\circ
\]
Then,
\[
60^\circ-x=48^\circ=\frac{4\pi}{15}
\]
and
\[
60^\circ+x=72^\circ=\frac{2\pi}{5}
\]
Therefore,
\[
\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{2\pi}{5}
=
\frac14\sin\frac{\pi}{5}
\]
Since,
\[
\sin\frac{3\pi}{10}
=
\sin\left(\frac{\pi}{2}-\frac{\pi}{5}\right)
=
\cos\frac{\pi}{5}
\]
Using exact value relation,
\[
\sin18^\circ\sin42^\circ\sin54^\circ=\frac18
\]
Hence,
\[
\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}
=
\frac18
\]
\[
\boxed{\frac18}
\]