Question Number 89202 by jagoll last updated on 16/Apr/20
$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\:=? \\ $$
Commented by jagoll last updated on 16/Apr/20
$${where}\:{sir}? \\ $$
Commented by jdmath last updated on 16/Apr/20
$${Dr}.\:{peyam}\:{video}\:{may}\:{be}\:{helpful} \\ $$
Commented by jdmath last updated on 16/Apr/20
$${youtube}…{i}\:{wasnt}\:{able}\:{to}\:{copy}\:{the}\:{link}..{search}\:{it} \\ $$
Commented by john santu last updated on 16/Apr/20
$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\prod}}\:\mathrm{sin}\:\left(\frac{\left({k}−\mathrm{1}\right)\pi}{{n}}\right)\:=\:\frac{{n}}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$${n}\:=\:\mathrm{7}\: \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right) \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{5}\pi}{\mathrm{7}}\right)\mathrm{sin}\:\left(\frac{\mathrm{6}\pi}{\mathrm{7}}\right)\:=\:\frac{\mathrm{7}}{\mathrm{64}}\:\:…\left({i}\right) \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{7}}\right)=\mathrm{sin}\:\left(\frac{\mathrm{6}\pi}{\mathrm{7}}\right) \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)=\mathrm{sin}\:\left(\frac{\mathrm{5}\pi}{\mathrm{7}}\right) \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)=\mathrm{sin}\:\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right) \\ $$$$\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{7}}\right)^{\mathrm{2}} \:=\:\frac{\mathrm{7}}{\mathrm{64}} \\ $$$$\Rightarrow\:\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{7}}\:=\:\sqrt{\frac{\mathrm{7}}{\mathrm{64}}} \\ $$$$=\:\frac{\sqrt{\mathrm{7}}}{\mathrm{8}} \\ $$