Задание №17129 ЕГЭ Профильной математике

$$\begin{gathered} a) 2{\cos }^2(x)-1-\sqrt{2}\cos \left(x\right)+1=0 \\ 2{\cos }^2(x)-\sqrt{2}\cos \left(x\right)=0 \\ \cos \left(x\right)(2\cos \left(x\right)-\sqrt{2})=0 \\ \left[\begin{array}{c}\cos \left(x\right)=0\\ \cos \left(x\right)=\raisebox{1ex}{$\sqrt{2}$}\!\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}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\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)-4\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. \\ -4\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{n}\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -4-\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. \\ -4.5\le \mathrm{n}\le -3 \\ \left\{\begin{array}{l}-4.5\le \mathrm{n}\le -3\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \left[\begin{array}{c}\mathrm{n} = -3\\ \mathrm{n} = -4\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.-4\mathrm{\pi }\end{array}\right. \\ \left[\begin{array}{c}\mathrm{x} = -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \mathrm{x} = -\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right. \\ \left(2\right) -4\mathrm{\pi }⩽\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}\le -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -4\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{n}\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -4-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le -\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ -\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \\ \left\{\begin{array}{l}-\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = -2 \\ \mathrm{x} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-4\mathrm{\pi } \\ \mathrm{x} = -\raisebox{1ex}{$15\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ -4\mathrm{\pi }⩽-\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}\le -\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -4\le \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{n}\le -\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -4+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ -\raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le -\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ -\raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \\ \left\{\begin{array}{l}-\raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le -\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = \varnothing \end{gathered}$$

$$\begin{gathered} Ответ: а) \{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}; \pm \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}\left\} \\ б\right) \{-\raisebox{1ex}{$15\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4; $}\right.-\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2; -{\displaystyle \raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$}\right.\} \end{gathered}$$