any short method available (Question:959695.0)

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SHOBHIT PANDEY

· started a discussion

· 1 Months ago

any short method available (Question:959695.0)

short

Question:

Find the value of $\frac{s i n π}{14} s i n \frac{3 π}{14} s i n \frac{5 π}{14} s i n \frac{7 π}{14} s i n \frac{9 π}{4} s i n \frac{11 π}{14} s i n \frac{13 π}{14}$

Options:

$\frac{1}{8}$

$\frac{1}{16}$

$\frac{1}{64}$

$\frac{1}{32}$

Solution:

$∴$ $s i n \frac{7 π}{14} = s i n \frac{π}{2} = 1$

$∵$ $s i n θ = c o s (\frac{π}{2} - θ)$

So, $s i n \frac{3 π}{14} = c o s (\frac{π}{2} - \frac{3 π}{14})$ = $c o s \frac{2 π}{7}$

Same as,

$s i n \frac{5 π}{14}$ = $c o s \frac{π}{7}$

$s i n \frac{9 π}{14} = c o s \frac{π}{7}$

$s i n \frac{11 π}{14} = c o s \frac{2 π}{7}$

$s i n \frac{13 π}{14} = s i n (\frac{14 π - π}{14}) = s i n (π - \frac{π}{14}) = s i n \frac{π}{14}$

$s i n \frac{π}{14} s i n \frac{3 π}{14} s i n \frac{5 π}{14} s i n \frac{7 π}{14} s i n \frac{9 π}{14} s i n \frac{11 π}{14} s i n \frac{13 π}{14}$

= $s i n \frac{π}{14} c o s \frac{2 π}{7} c o s \frac{π}{7} c o s \frac{π}{7} c o s \frac{2 π}{7} s i n \frac{π}{14}$

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any short method available (Question:959695.0)

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