Question Number 179659 by mathlove last updated on 31/Oct/22
$${cos}\frac{\mathrm{2}\pi}{\mathrm{7}}=\frac{{ab}}{{c}} \\ $$$${cos}\frac{\mathrm{4}\pi}{\mathrm{7}}=\frac{{bc}}{{a}} \\ $$$${cos}\frac{\mathrm{6}\pi}{\mathrm{7}}=\frac{{ac}}{{b}}\:\:\:\:\:\:\: \\ $$$${faind}\:\:\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =? \\ $$
Answered by DvMc last updated on 31/Oct/22
$${cos}\frac{\mathrm{2}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{6}\pi}{\mathrm{7}}=\frac{{ab}}{{c}}×\frac{{ac}}{{b}}={a}^{\mathrm{2}} \\ $$$${cos}\frac{\mathrm{4}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{2}\pi}{\mathrm{7}}={b}^{\mathrm{2}} \\ $$$${cos}\frac{\mathrm{6}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{4}\pi}{\mathrm{7}}={c}^{\mathrm{2}} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={cos}\frac{\mathrm{2}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{6}\pi}{\mathrm{7}}+{cos}\frac{\mathrm{4}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{2}\pi}{\mathrm{7}}+{cos}\frac{\mathrm{6}\pi}{\mathrm{7}}\:{cos}\frac{\mathrm{4}\pi}{\mathrm{7}} \\ $$
Commented by MJS_new last updated on 01/Nov/22
$$=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$