The Value of \( \sin\frac{\pi}{10}\sin\frac{13\pi}{10} \)
Question
Find the value of
\[ \sin\frac{\pi}{10}\sin\frac{13\pi}{10} \]
(a) \(\frac12\)
(b) \(-\frac12\)
(c) \(-\frac14\)
(d) \(1\)
Solution
Use the identity
\[ \sin(\theta+\pi)=-\sin\theta \]
Since
\[ \frac{13\pi}{10} = \pi+\frac{3\pi}{10}, \]
we have
\[ \sin\frac{13\pi}{10} = -\sin\frac{3\pi}{10} \]
Therefore,
\[ \sin\frac{\pi}{10}\sin\frac{13\pi}{10} = -\sin\frac{\pi}{10}\sin\frac{3\pi}{10} \]
Now use the exact values
\[ \sin18^\circ=\frac{\sqrt5-1}{4}, \qquad \sin54^\circ=\frac{\sqrt5+1}{4} \]
Hence,
\[ -\sin18^\circ\sin54^\circ = -\frac{(\sqrt5-1)(\sqrt5+1)}{16} \]
\[ = -\frac{5-1}{16} \]
\[ = -\frac{4}{16} = -\frac14 \]
Final Answer
\[ \boxed{-\frac14} \]
Hence, the correct option is (c) \(-\frac14\).