11th Class Mathematics Trigonometric Identities Question Bank Critical Thinking

  • question_answer 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 [IIT 1991; MNR 1992]

    A) \[\frac{1}{8}\]

    B) \[\frac{1}{16}\]

    C) \[\frac{1}{32}\]

    D) \[\frac{1}{64}\]

    Correct Answer: D

    Solution :

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


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