Question Number 146310 by mathdanisur last updated on 12/Jul/21
$$\mathrm{4}\:{sin}\left({x}\right)\:{cos}\left({x}\right)\:\geqslant\:\sqrt{\mathrm{3}} \\ $$
Commented by mathdanisur last updated on 12/Jul/21
$${Solve}\:{the}\:{trigonometric}\:{inequality} \\ $$
Commented by MJS_new last updated on 12/Jul/21
$$\mathrm{let}\:{a}_{{k}} =\frac{\pi}{\mathrm{6}}+{k}\pi\wedge{b}_{{k}} =\frac{\pi}{\mathrm{3}}+{k}\pi\wedge{k}\in\mathbb{Z} \\ $$$$\Rightarrow \\ $$$$\mathrm{4sin}\:{x}\:\mathrm{cos}\:{x}\:\geqslant\sqrt{\mathrm{3}}\:\Leftrightarrow\:{a}_{{k}} \leqslant{x}\leqslant{b}_{{k}} \:\forall{k}\in\mathbb{Z} \\ $$
Commented by mathdanisur last updated on 13/Jul/21
$${thanks}\:{Ser} \\ $$
Answered by puissant last updated on 12/Jul/21
$$\mathrm{4sin}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{x}\right)=\mathrm{2sin}\left(\mathrm{2x}\right) \\ $$$$\Rightarrow\mathrm{2sin}\left(\mathrm{2x}\right)\geqslant\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{sin}\left(\mathrm{2x}\right)\geqslant\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{sin}\frac{\pi}{\mathrm{3}}.. \\ $$