If K= ( (π)/(18)) ((5 π)/(18)) ((7 π)/(18)) , then what is the value of K ?

Correct answer: C. (1)/(8) K= ((π)/(18)) ((5 π)/(18)) ((7 π)/(18)) We know, 2 A B= (A-B)- (A+B) aligned & K=(1)/(2) · (π)/(18) [2 (5 π)/(18) (7 π)/(18)] & =(1)/(2) · (π)/(18) [ (2 π)/(18)- (12 π)/(18)] & =(1)/(2) · (1)/(2) [2 (π)/(18) (2 π)/(18)- 2 (π)/(18) (2 π)/(3)] & =(1)/(4)[ ((3 π)/(18))+ ((-π)/(18))-2 (π)/(18) (π-(π)/(3))] & =(1)/(4) [ (π)/(6)- (π)/(18)+2 (π)/(18) · (π)/(3)] & =(1)/(4) [ (π)/(6)- (π)/(18)+2 (π)/(18) · (1)/(2)] & =(1)/(4) (π)/(6)=(1)/(4) × (1)/(2)=(1)/(8) aligned

NDA 2017

Exam: NDA
Year: 2017
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NDA 2017 · Mathematics · Trigonometric Ratios & Identities

If K= ( (π)/(18)) ((5 π)/(18)) ((7 π)/(18)) , then what is the...

K=\sin \left(

\frac{\pi}{18}\right) \sin\left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right)

,

then what is the value of K ?$$

Options

A $\frac{1}{2}$
B $\frac{1}{4}$
C $\frac{1}{8}$
D $\frac{1}{16}$

Correct answer

C. $\frac{1}{8}$

Step-by-step solution

\mathrm{K}=\sin \left(\frac{\pi}{18}\right) \sin\left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right) We know, 2 \sin \mathrm{ A} \sin \mathrm{ B}=\cos (\mathrm{A}-\mathrm{B})-\cos (\mathrm{A}+\mathrm{B})

\begin{aligned}& \mathrm{K}=\frac{1}{2} \cdot \sin \frac{\pi}{18} \left[2 \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18}\right]\\ & =\frac{1}{2} \cdot \sin \frac{\pi}{18} \left[\cos \frac{2 \pi}{18}-\cos \frac{12 \pi}{18}\right]\\ & =\frac{1}{2} \cdot \frac{1}{2} \left[2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{18}- 2 \sin \frac{\pi}{18} \cos \frac{2 \pi}{3}\right]\\ & =\frac{1}{4}\left[\sin \left(\frac{3 \pi}{18}\right)+\sin\left(\frac{-\pi}{18}\right)-2 \sin \frac{\pi}{18} \cos \left(\pi-\frac{\pi}{3}\right)\right]\\ & =\frac{1}{4} \left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+2 \sin \frac{\pi}{18} \cdot \cos \frac{\pi}{3}\right]\\ & =\frac{1}{4} \left[\sin \frac{\pi}{6}-\sin \frac{\pi}{18}+2 \sin \frac{\pi}{18} \cdot \frac{1}{2}\right]\\ & =\frac{1}{4} \sin \frac{\pi}{6}=\frac{1}{4} \times \frac{1}{2}=\frac{1}{8} \end{aligned}

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