If $sin\left( \frac{\pi}{18} \right)sin\left( \frac{5\pi}{18} \right)sin\left( \frac{7\pi}{18} \righ

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If $sin\left( \frac{\pi}{18} ight)sin\left( \frac{5\pi}{18} ight)sin\left( \frac{7\pi}{18} ight) = K$, then the value of $sin\left( \frac{10	ext{ }K\pi}{3} ight)$ is :

Accepted Answer

Sol. [Key Concept: Use $sin	heta\ sin(60 - 	heta)\ sin(60 + 	heta) = \frac{14}{\sin}\ 3	heta$ to get $K = 1/8$, then evaluate.]$$K = sin10^{\circ}sin50^{\circ}sin70^{\circ}$$$$\begin{matrix} & \ = \frac{1}{4}sin30^{\circ} = \frac{1}{8} \ & sin10K\frac{\pi}{3} = sin\left( 10 	imes \frac{1}{8} \cdot \frac{\pi}{3} ight) = sin\frac{5\pi}{12} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \end{matrix}$$

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