Question

Find the value of below expression :—
$\underline{ \boxed{ \sin \left( \frac{\pi}{14} \right) \sin \left( \frac{3\pi}{14} \right) \sin \left( \frac{5\pi}{14} \right) }}$

☺ give step-by-step explanation

☺ class 11

☺ Topic — Trigonometry

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Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

$\blacksquare \:$TO DETERMINE

$\displaystyle \: sin \: \frac{\pi}{14} \: \: sin \: \frac{3\pi}{14} \: \: sin \: \frac{5\pi}{14}$

$\blacksquare \: \:$CALCULATION

$\displaystyle \: sin \: \frac{3\pi}{14} \: = sin \: ( \frac{\pi}{2} - \frac{4\pi}{14} \: ) = cos\: \frac{4\pi}{14}$

Also

$\displaystyle \: sin \: \frac{5\pi}{14} \: = sin \: ( \frac{\pi}{2} - \frac{2\pi}{14} \: ) = cos\: \frac{2\pi}{14}$

Let

$\displaystyle \:x = \: \frac{\pi}{14}$

Then

$\displaystyle \: sin \: \frac{\pi}{14} \: \: sin \: \frac{3\pi}{14} \: \: sin \: \frac{5\pi}{14}$

$= \displaystyle \: sin x\: \: cos 4x\: \: cos2x$

$= \displaystyle \: sin x\: \: cos 2x\: \: cos4x$

$= \displaystyle \: \frac{1}{2cosx}( \:2cosx sin x)\: \: cos </u></em></strong><strong><em><u>2</u></em></strong><strong><em><u>x\: \: </u></em></strong><strong><em><u>cos4</u></em></strong><strong><em><u>x$

$= \displaystyle \: \frac{1}{2cosx}\: sin 2x\: \: cos </u></em></strong><strong><em><u>2</u></em></strong><strong><em><u>x\: \: </u></em></strong><strong><em><u>cos4</u></em></strong><strong><em><u>x$

$= \displaystyle \: \frac{1}{4cosx}\: (2sin 2x\: cos 2x)\: \: cos4x$

$= \displaystyle \: \frac{1}{4cosx}\: sin 4x\: \: cos 4x$

$= \displaystyle \: \frac{1}{8cosx}\: (2sin 4x\: \: cos 4x)$

$= \displaystyle \: \frac{1}{8cosx}\: sin 8x$

$= \displaystyle \: \frac{1}{8cos \frac{\pi}{14} }\: sin \frac{ 8\pi}{14}$

$= \displaystyle \: \frac{1}{8cos \frac{\pi}{14} }\: sin( \frac{\pi}{2} + \frac{ \pi}{14} )$

$= \displaystyle \: \frac{1}{8cos \frac{\pi}{14} }\: cos \frac{ \pi}{14}$

$= \displaystyle \: \frac{1}{8}$

Answer 2

Answer:

Find the value of below expression :—

$\underline{ \boxed{ \sin \left( \frac{\pi}{14} \right) \sin \left( \frac{3\pi}{14} \right) \sin \left( \frac{5\pi}{14} \right) }}$

☺ give step-by-step explanation

☺ class 11

☺ Topic — Trigonometry

☺ #no spam #please