Задание №17125 ЕГЭ Профильной математике

 $$\begin{gathered} a) 2{\cos }^2\left(x\right)=-\sqrt{3}\cos \left(x\right) \\ 2{\cos }^2\left(x\right)+\sqrt{3}\cos \left(x\right)=0 \\ \cos \left(x\right)(2\cos \left(x\right)+\sqrt{3})=0 \\ \left[\begin{array}{c}\cos \left(x\right)=0\\ \cos \left(x\right) = -\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right. \\ \left[\begin{array}{c}x = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n} \left(1\right)\\ x = \pm \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{\pi n} \left(2\right)\end{array}\right. \\ б)\mathrm{Найдем} \mathrm{корни} \mathrm{уравнения} \mathrm{с} \mathrm{помощью} \mathrm{двойного} \mathrm{неравенства}: \\ \left(1\right)\mathrm{\pi }⩽\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n}\le \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ 1\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ 1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{n}\le 2 \\ \left\{\begin{array}{l}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{n}\le 2\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \left[\begin{array}{c}\mathrm{n} = 1\\ \mathrm{n} = 2\end{array}\right. \\ \left[\begin{array}{c}\mathrm{x} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi }\\ \mathrm{x} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+2\mathrm{\pi }\end{array}\right. \\ \left[\begin{array}{c}\mathrm{x} = \raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \mathrm{x} = \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right. \\ \left(2\right) \mathrm{\pi }⩽\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{\pi n}\le \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ 1\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ \left\{\begin{array}{l}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} \in \varnothing \\ \mathrm{\pi }⩽-\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{\pi n}\le \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ 1\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ 1+\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ \left\{\begin{array}{l}\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = 1 \\ x = -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{\pi } \\ \mathrm{x} = \raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ \mathrm{Ответ}: \mathrm{а}) \{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}; \pm \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.+2\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}\left\} \\ \mathrm{б}\right) \{\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.; \raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.; \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\} \end{gathered}$$