Задание №17108 ЕГЭ Профильной математике

 $$\begin{gathered} a) 2{\cos }^2\left(x\right)-1+\cos \left(x\right)+1=0 \\ \cos \left(x\right)(2\cos \left(x\right)+1)=0 \\ \left[\begin{array}{c}\cos \left(x\right)=0\\ \cos \left(x\right)=-\raisebox{1ex}{$1$}\!\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}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\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)-\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⩽\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n}\le -\mathrm{\pi } \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{n}\le -1 \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{n}\le -1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -3\le \mathrm{n}\le -1.5 \\ \left\{\begin{array}{l}-3\le \mathrm{n}\le -1.5\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \left[\begin{array}{c}\mathrm{n} = -3\\ \mathrm{n} = -2\end{array}\right. \\ \left[\begin{array}{c}\mathrm{x} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-3\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}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{\pi }\\ \mathrm{x} = -\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{\pi }\end{array}\right. \\ \left(2\right) -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⩽\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+2\mathrm{\pi n}\le -\mathrm{\pi } \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+2\mathrm{n}\le -1 \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\le 2\mathrm{n}\le -1-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ -\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\le 2\mathrm{n}\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ -\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ \left\{\begin{array}{l}-\raisebox{1ex}{$19$}\!\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} = -1 \\ \mathrm{x} = \raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.-2\mathrm{\pi } \\ \mathrm{x} = -\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\mathrm{\pi } \\ -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.⩽-\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+2\mathrm{\pi n}\le -\mathrm{\pi } \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le -\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+2\mathrm{n}\le -1 \\ -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\le 2\mathrm{n}\le -1+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ -\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\le 2\mathrm{n}\le -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \\ -\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right. \\ \left\{\begin{array}{l}-\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = \varnothing \end{gathered}$$

$$\begin{gathered} Ответ: а) \{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}; \pm \raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+2\mathrm{\pi n}, \mathrm{n}\in \mathrm{\mathbb{N}}\left\} \\ б\right) \{-\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{\pi }; -\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\mathrm{\pi }; -\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\mathrm{\pi }\} \end{gathered}$$