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Sum of Sines of Angles in Arithmetic Progression

sinπ14sin3π14...

Question

$sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$ =_________?

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Solution

$sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$
$= sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{π}{2} sin (π - \frac{5 π}{14}) sin (π - \frac{3 π}{14}) sin (π - \frac{π}{14})$
$= {(sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})}^{2}$
$= {[cos (\frac{π}{2} - \frac{π}{14}) cos (\frac{π}{2} - \frac{3 π}{14}) cos (\frac{π}{2} - \frac{5 π}{14})]}^{2}$
$= {(cos \frac{3 π}{7} cos \frac{2 π}{7} cos \frac{π}{7})}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{2 sin \frac{π}{7}} \times 2 sin \frac{π}{7} cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{2 sin \frac{π}{7}} \times sin \frac{2 π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{4 sin \frac{π}{7}} \times 2 sin \frac{2 π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{8 sin \frac{π}{7}} \times 2 sin \frac{4 π}{7} cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{8 sin \frac{π}{7}} \times 2 sin (π - \frac{3 π}{7}) cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{8 sin \frac{π}{7}} \times 2 sin \frac{3 π}{7} cos \frac{3 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{8 sin \frac{π}{7}} \times sin \frac{6 π}{7} ⎤ ⎥ ⎥ ⎦}^{2}$
$= {⎡ ⎢ ⎢ ⎣ \frac{1}{8 sin \frac{π}{7}} \times sin (π - \frac{π}{7}) ⎤ ⎥ ⎥ ⎦}^{2}$
$= \frac{1}{64}$

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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}

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},

k

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},

k

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Sin π/14 sin 3π/14 sin 5π/14 sin 7π/14 sin 9π/14 sin 11π/14 sin 13π/14 sin 13π/14 = ? ( Also Name the trigonometric functions used)

Q. The value of

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}

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