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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
$\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}=$
  • A
    $0$
  • B
    $\frac{1}{2}$
  • C
    $\frac{1}{4}$
  • ✓
    $-\frac{1}{8}$
✓

Answer

Correct option: D.
$-\frac{1}{8}$
(D)
$\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$
$\cos \alpha \cdot \cos 2 \alpha \cdot \cos 2^2 \alpha \cdot \cos 2^3 \alpha \ldots . . \cos 2^{n-1} \alpha$
$=\frac{\sin 2^n \alpha}{2^n \sin \alpha}, \text { if } \alpha \neq n \pi$
$=1, \text { if } \alpha=2 n \pi$
$=-1, \text { if } \alpha=(2 n+1) \pi$
$=\left[\frac{\sin \left(2^3 \cdot \frac{\pi}{7}\right)}{2^3 \sin \left(\frac{\pi}{7}\right)}\right]$
$=\frac{\sin \frac{8 \pi}{7}}{8 \sin \frac{\pi}{7}}$
$=-\frac{1}{8} \quad \ldots\left[\because \sin \frac{8 \pi}{7}=\sin \left(\pi+\frac{\pi}{7}\right)=-\sin \frac{\pi}{7}\right]$

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