Question

Tentuka nilai dari [tex]\displaystyle \sin \frac{\pi}{14}\sin \frac{3\pi}{14}\sin \frac{5\pi}{14}\sin \frac{7\pi}{14}\sin \frac{9\pi}{14}\sin \frac{11\pi}{14}\sin \frac{13\pi}{14}[/tex]

Answer 1

Verified answer

Trigonometri

sin (π/2 - x) = cos x

sin 2x = 2 sin x cos x

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π/14 = a

sin (3π/7) = sin (π/2 - 4π/14) = cos (4π/14)

sin (5π/7) = sin (π/2 - 2π/14) = cos (2π/14)

sin (π/14) sin (3π/4) sin (5π/14)

= sin (π/14) cos (4π/14) cos (2π/14)

= sin a cos 2a cos 4a

= sin 2a cos 2a cos 4a / (2 cos a)

= sin 4a cos 4a / (4 cos a)

= sin 8a / (8 sin (7a + a))

= 1/8

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sin (7π/14) = 1

sin (9π/14) = sin (π - 5π/14) = sin (5π/14)

sin (11π/14) = sin (3π/14)

sin (13π/14) = sin (π/14)

•

[tex]\displaystyle \sin \frac{\pi}{14}\sin \frac{3\pi}{14}\sin \frac{5\pi}{14}\sin \frac{7\pi}{14}\sin \frac{9\pi}{14}\sin \frac{11\pi}{14}\sin \frac{13\pi}{14} \\ \\ = \: {(\displaystyle \sin \frac{\pi}{14}\sin \frac{3\pi}{14}\sin \frac{5\pi}{14})}^{2} \\ \\ = \: {( \frac{1}{8}) }^{2} \: = \: \frac{1}{64} [/tex]