If k = sinπ18.sin5π18.sin7π18, then the numerical value of k

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If k = sin $\frac{π}{18}$ .sin $\frac{5 π}{18}$ .sin $\frac{7 π}{18}$ , then the numerical value of k is

Options:

A .

\frac{1}{4}

B .

\frac{1}{8}

C .

\frac{1}{16}

D . None of these

Answer: Option B
:
B

(b) We have k = sin $\frac{π}{18}$ .sin $\frac{5 π}{18}$ .sin $\frac{7 π}{18}$
= cos $(\frac{π}{2} - \frac{π}{18})$ cos $(\frac{π}{2} - \frac{5 π}{18})$ cos $(\frac{π}{2} - \frac{7 π}{18})$
= cos $\frac{π}{9}$ cos $\frac{2 π}{9}$ cos $\frac{4 π}{9}$ = $\frac{s i n 2^{3} \frac{π}{9}}{2^{3} s i n \frac{π}{9}}$ = $\frac{s i n \frac{8 π}{9}}{8 s i n \frac{π}{9}}$
= $\frac{s i n (π - \frac{π}{9})}{8 s i n \frac{π}{9}}$ = $\frac{1}{8}$

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