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  • Fill in the blanks If k=sin⁡(π/18) sin⁡(5π/18) sin⁡(7π/18) then the numerical value of k is _________.

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If k=\sin \left( \frac{ \pi }{18} \right) \sin \left( \frac{5 \pi }{18} \right) \sin \left( \frac{7 \pi }{18} \right) \\\\then the numerical value of k is

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\\ k=\sin \left( \frac{ \pi }{18} \right) \sin \left( \frac{5 \pi }{18} \right) \sin \left( \frac{7 \pi }{18} \right) \\\\ k=\sin 10\sin 50\sin 70 \\\\ k=\sin 10\cos 40\cos 20 \\\\ k=\sin 10\frac{1}{2} \times \left[ 2\cos 40\cos 20 \right] =\sin 10\frac{1}{2} \left[ \cos 60+\cos 20 \right] \\\\ k=\frac{1}{2}\sin 10 \left[ \frac{1}{2}+\cos 20 \right] \\\\ k=\frac{1}{4}\sin 10+\frac{1}{2}\sin 10\cos 20 \\\\ k=\frac{1}{4}\sin 10+\frac{1}{4} \left[ \sin 30 - \sin 10 \right] \\\\ k=\frac{1}{4}\sin 30=\frac{1}{4} \times \frac{1}{2}=\frac{1}{8} \\\\

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