Question Number 119506 by bemath last updated on 25/Oct/20
$$\:\:\:\mathrm{cos}\:\frac{\pi}{\mathrm{5}}\mathrm{cos}\:{x}+\mathrm{sin}\:\frac{\pi}{\mathrm{5}}\mathrm{sin}\:{x}\:\leqslant\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$
Answered by 1549442205PVT last updated on 25/Oct/20
$$\:\:\:\mathrm{cos}\:\frac{\pi}{\mathrm{5}}\mathrm{cos}\:{x}+\mathrm{sin}\:\frac{\pi}{\mathrm{5}}\mathrm{sin}\:{x}\:\leqslant\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\Leftrightarrow\mathrm{cos}\left(\mathrm{x}−\frac{\pi}{\mathrm{5}}\right)\leqslant\mathrm{cos}\frac{\pi}{\mathrm{4}} \\ $$$$\Leftrightarrow\mathrm{x}−\frac{\pi}{\mathrm{5}}\geqslant\frac{\pi}{\mathrm{4}}+\mathrm{2m}\pi \\ $$$$\Leftrightarrow\frac{\boldsymbol{\pi}}{\mathrm{4}}\:+\mathrm{2}\boldsymbol{\mathrm{m}\pi}\leqslant\boldsymbol{\mathrm{x}}\leqslant\frac{\mathrm{7}\boldsymbol{\pi}}{\mathrm{4}}+\mathrm{2}\boldsymbol{\mathrm{m}\pi} \\ $$