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Question Number 197073 by Abdullahrussell last updated on 07/Sep/23

Answered by Frix last updated on 07/Sep/23

sin (π/7) sin ((2π)/7) sin ((3π)/7) =x Use trigonometric formulas to get (1/4)(cos ((3π)/(14)) +cos (π/(14)) −sin (π/7))=x (1) Now comes the “trick” (cos ((3π)/(14)) +cos (π/(14)) −sin (π/7))^3 =64x^3 Again use trigonometric formulas to get (7/4)(cos ((3π)/(14)) +cos (π/(14)) −sin (π/7))=64x^3 (2) ⇒ 4x=((256x^3 )/7) we know x>0 ⇒ x=((√7)/8)

$$\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\:\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{7}}\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{7}}\:={x} \\ $$$$\mathrm{Use}\:\mathrm{trigonometric}\:\mathrm{formulas}\:\mathrm{to}\:\mathrm{get} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:+\mathrm{cos}\:\frac{\pi}{\mathrm{14}}\:−\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)={x}\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Now}\:\mathrm{comes}\:\mathrm{the}\:``\mathrm{trick}'' \\ $$$$\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:+\mathrm{cos}\:\frac{\pi}{\mathrm{14}}\:−\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)^{\mathrm{3}} =\mathrm{64}{x}^{\mathrm{3}} \\ $$$$\mathrm{Again}\:\mathrm{use}\:\mathrm{trigonometric}\:\mathrm{formulas}\:\mathrm{to}\:\mathrm{get} \\ $$$$\frac{\mathrm{7}}{\mathrm{4}}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:+\mathrm{cos}\:\frac{\pi}{\mathrm{14}}\:−\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)=\mathrm{64}{x}^{\mathrm{3}} \:\left(\mathrm{2}\right) \\ $$$$\Rightarrow \\ $$$$\mathrm{4}{x}=\frac{\mathrm{256}{x}^{\mathrm{3}} }{\mathrm{7}} \\ $$$$\mathrm{we}\:\mathrm{know}\:{x}>\mathrm{0}\:\Rightarrow\:{x}=\frac{\sqrt{\mathrm{7}}}{\mathrm{8}} \\ $$

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