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Question Number 90628 by Josephbaraka@gmail.com last updated on 25/Apr/20

The value of sin (π/(14)) sin ((3π)/(14)) sin ((5π)/(14)) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sin}\:\frac{\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{5}\pi}{\mathrm{14}}\:\:\mathrm{is} \\ $$

Commented by jagoll last updated on 25/Apr/20

x = (π/(14)) ⇒7x = (π/2) w = sin x sin 3x sin 5x 2w cos x = sin 2x sin 5x sin 3x 4w cos x = (cos 3x−cos 7x) sin 3x 4w cos x = cos 3x sin 3x 8w cos x = sin 6x w = ((sin 6x)/(8cos x)) = ((cos ((π/(14))))/(cos ((π/(14))))) = (1/8) sin 6x = sin (((3π)/7)) = cos ((π/(14)))

$${x}\:=\:\frac{\pi}{\mathrm{14}}\:\Rightarrow\mathrm{7}{x}\:=\:\frac{\pi}{\mathrm{2}} \\ $$$${w}\:=\:\mathrm{sin}\:{x}\:\mathrm{sin}\:\mathrm{3}{x}\:\mathrm{sin}\:\mathrm{5}{x} \\ $$$$\mathrm{2}{w}\:\mathrm{cos}\:{x}\:=\:\mathrm{sin}\:\mathrm{2}{x}\:\mathrm{sin}\:\mathrm{5}{x}\:\mathrm{sin}\:\mathrm{3}{x} \\ $$$$\mathrm{4}{w}\:\mathrm{cos}\:{x}\:=\:\left(\mathrm{cos}\:\mathrm{3}{x}−\mathrm{cos}\:\mathrm{7}{x}\right)\:\mathrm{sin}\:\mathrm{3}{x} \\ $$$$\mathrm{4}{w}\:\mathrm{cos}\:{x}\:=\:\mathrm{cos}\:\mathrm{3}{x}\:\mathrm{sin}\:\mathrm{3}{x} \\ $$$$\mathrm{8}{w}\:\mathrm{cos}\:{x}\:=\:\mathrm{sin}\:\mathrm{6}{x} \\ $$$${w}\:=\:\frac{\mathrm{sin}\:\mathrm{6}{x}}{\mathrm{8cos}\:{x}}\:=\:\frac{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right)}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\mathrm{sin}\:\mathrm{6}{x}\:=\:\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:=\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right) \\ $$$$ \\ $$

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