Question

The value of $ \sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14} $ is equal to

• A. $ \frac{1}{8} $
• B. $ \frac{1}{16} $
• C. $ \frac{1}{32} $
• D. $ \frac{1}{64} $

Answer 1

Answer: (D)

$ \sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14} $
$ =\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\times 1 $
$ \times \sin \left( \pi -\frac{5\pi }{14} \right)\sin \left( \pi -\frac{3\pi }{14} \right)\sin \left( \pi -\frac{\pi }{14} \right) $
$ ={{\left[ \sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14} \right]}^{2}}=\frac{1}{64} $