If sin-14sin3-14sin5-14sin7-14sin9-14sin11-14sin13-14-1k- then k is equal to

Sinπ14sin3π14sin5π14sin7π14sin9π14sin11π14sin13π14 = sinπ14sin3π14sin5π14sinπ2sin(π 5π14) sin(π 3π14) sin(π π14) = sin^2π14sin^2 3π4sin^25π14 = (sinπ14sin3π ...

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Standard XIII

Mathematics

Sum of Sines of Angles in Arithmetic Progression

If sinπ14sin3...

Question

If $sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14} = \frac{1}{k},$ then $k$ is equal to

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Solution

$sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$
$= sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{π}{2} sin (π - \frac{5 π}{14}) sin (π - \frac{3 π}{14}) sin (π - \frac{π}{14})$
$= {sin}^{2} \frac{π}{14} {sin}^{2} \frac{3 π}{4} {sin}^{2} \frac{5 π}{14}$
$= {(sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})}^{2}$
$= {[cos (\frac{π}{2} - \frac{π}{14}) cos (\frac{π}{2} - \frac{3 π}{14}) cos (\frac{π}{2} - \frac{5 π}{14})]}^{2}$
$= {[cos \frac{3 π}{7} cos \frac{2 π}{7} cos \frac{π}{7}]}^{2} = {[cos (π - \frac{4 π}{7}) cos \frac{2 π}{7} cos \frac{π}{7}]}^{2} = {[- cos \frac{2 π}{7} cos \frac{π}{7} cos \frac{4 π}{7}]}^{2} = {[cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{4 π}{7}]}^{2} = {⎡ ⎢ ⎢ ⎢ ⎣ \frac{sin \frac{8 π}{7}}{2^{3} sin \frac{π}{7}} ⎤ ⎥ ⎥ ⎥ ⎦}^{2} = \frac{1}{64} {⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{sin (π + \frac{π}{7})}{sin \frac{π}{7}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}^{2} = \frac{1}{64} ∴ k = 64$

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Similar questions

Q. If

sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14} = \frac{1}{k},

then

k

is equal to

Q. If

sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14} = \frac{1}{k},

then

k

is equal to

Q. The value of

sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}

is equal to

sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}

is equal to

Q. If the value of

sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}

is equal to

k^{2}

, then

k^{2}

is equal to

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Sum of Sines of Angles in Arithmetic Progression

Standard XIII Mathematics

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If sinπ14sin3π14sin5π14sin7π14sin9π14sin11π14sin13π14=1k, then k is equal to

If $sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14} = \frac{1}{k},$ then $k$ is equal to