Question Number 75040 by mathocean1 last updated on 06/Dec/19
$$\mathrm{Please}\:\mathrm{can}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\: \\ $$$$\mathrm{to}\:\mathrm{show}\:\mathrm{that}: \\ $$$$\mathrm{cos}\:\frac{\mathrm{47}\Pi}{\mathrm{13}}=\mathrm{sin}\:\frac{\mathrm{23}\Pi}{\mathrm{26}}=\mathrm{sin}\frac{\mathrm{3}\Pi}{\mathrm{26}} \\ $$
Answered by Kunal12588 last updated on 06/Dec/19
$${cos}\frac{\mathrm{47}\pi}{\mathrm{13}}={cos}\left(\frac{\mathrm{52}\pi−\mathrm{5}\pi}{\mathrm{13}}\right)={cos}\left(\mathrm{4}\pi−\frac{\mathrm{5}\pi}{\mathrm{13}}\right) \\ $$$$={cos}\left(\frac{\mathrm{5}\pi}{\mathrm{13}}\right)={sin}\left(\frac{\pi}{\mathrm{2}}+\frac{\mathrm{5}\pi}{\mathrm{13}}\right)={sin}\frac{\mathrm{23}\pi}{\mathrm{26}} \\ $$$$={sin}\frac{\mathrm{26}\pi−\mathrm{3}\pi}{\mathrm{26}}={sin}\left(\pi−\frac{\mathrm{3}\pi}{\mathrm{26}}\right)={sin}\frac{\mathrm{3}\pi}{\mathrm{26}} \\ $$
Commented by mathocean1 last updated on 07/Dec/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$