Find the value of sin-14-sin3-14-sin5-14-sin7-14sin9-14sin11-14-sin13-14-

The correct option is C 164 Let a=sinπ #160;/ 14 #160;sin 3π #160;/ 14 #160;sin 5π #160;/ 14 #160;sin 7π #160;/ 14 #160;sin 9π ...

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Byju's Answer

Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

Find the valu...

Question

Find the value of $sin \frac{π}{14} . sin \frac{3 π}{14} . sin \frac{5 π}{14} . sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} . sin \frac{13 π}{14}$ .

\frac{1}{64}

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\frac{3}{64}

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\frac{5}{64}

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\frac{7}{64}

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Solution

The correct option is C $\frac{1}{64}$
Let $a = sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$
$\Rightarrow a = sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{π}{2} sin (π - \frac{5 π}{14}) sin (π - \frac{3 π}{14}) sin (π - \frac{π}{14})$
$\Rightarrow a = {(sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})}^{2}$ $[∵ sin (π - x) = sin x]$
$a = {⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{(2 sin \frac{π}{14} cos \frac{π}{14}) (2 sin \frac{3 π}{14} cos \frac{3 π}{14}) (2 sin \frac{5 π}{14} cos \frac{5 π}{14})}{8 cos \frac{π}{14} cos \frac{3 π}{14} cos \frac{5 π}{14}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}^{2}$
$\Rightarrow a = {⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{sin \frac{π}{7} sin \frac{3 π}{7} sin \frac{5 π}{7}}{8 sin (\frac{π}{2} - \frac{π}{14}) sin (\frac{π}{2} + \frac{3 π}{14}) sin (\frac{π}{2} - \frac{5 π}{14})} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}^{2} {∵ 2 sin θ cos θ = sin 2 θ}$
$\Rightarrow a = {⎛ ⎜ ⎜ ⎜ ⎝ \frac{sin \frac{π}{7} sin \frac{3 π}{7} sin \frac{5 π}{7}}{8 sin \frac{3 π}{7} sin \frac{5 π}{7} sin \frac{π}{7}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2} = {(\frac{1}{8})}^{2} = \frac{1}{64}$

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Similar questions

Q. The value of

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

is equal to

Q. The Value of

sin \frac{π}{14} \cdot sin \frac{3 π}{14} \cdot sin \frac{5 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}

=_________?

$c o s \frac{π}{7}$ . $c o s \frac{3 π}{7}$ . $c o s \frac{5 π}{7}$ are the roots of the equation $8 x^{3} - 4 x^{2} - 4 x + 1 = 0$ . Then the value of $s i n \frac{π}{14} . s i n \frac{3 π}{14} . s i n \frac{5 π}{14}$ is

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

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Find the value of sinπ14.sin3π14.sin5π14.sin7π14sin9π14sin11π14.sin13π14.

Find the value of $sin \frac{π}{14} . sin \frac{3 π}{14} . sin \frac{5 π}{14} . sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} . sin \frac{13 π}{14}$ .