Задание №17131 ЕГЭ Профильной математике

$$\begin{gathered} a) 1-2{\sin }^2\left(x\right)-\sqrt{2}\sin \left(x\right)-1=0 \\ 2{\sin }^2(x)+\sqrt{2}\sin \left(x\right)=0 \\ \sin \left(x\right)(2\sin \left(x\right)+\sqrt{2})=0 \\ \left[\begin{array}{c}\sin \left(x\right)=0\\ \sin \left(x\right)=-\raisebox{1ex}{$\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right. \\ \left[\begin{array}{c}x=\mathrm{\pi n} \left(1\right)\\ x=-\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n} \left(2\right)\\ x = -\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n} \left(3\right)\end{array}\right. \\ б) \mathrm{Найдем} \mathrm{корни} \mathrm{уравнения} \mathrm{с} \mathrm{помощью} \mathrm{двойного} \mathrm{неравенства}: \\ \left(1\right)\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \mathrm{\pi n}\le 3\mathrm{\pi } \\ 1.5\le \mathrm{n}\le 3 \\ \left\{\begin{array}{l}1.5\le \mathrm{n}\le 3\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \left[\begin{array}{c}\mathrm{n} = 2\\ \mathrm{n} = 3\end{array}\right. \\ \left[\begin{array}{c}\mathrm{x} = 2\mathrm{\pi }\\ \mathrm{x} = 3\mathrm{\pi }\end{array}\right. \\ \left(2\right) \raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le -\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}\le 3\mathrm{\pi } \\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2n\le 3 \\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2n\le 3+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \\ \left\{\begin{array}{l}\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = 1 \\ \mathrm{x} = -\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi } \\ \mathrm{x} = \raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \raisebox{1ex}{$\left(3\right) 3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le -3\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}\le 3\mathrm{\pi } \\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le -\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2n\le 3 \\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2n\le 3+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\le 2\mathrm{n}\le \raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le \raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \\ \left\{\begin{array}{l}\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\le \mathrm{n}\le \raisebox{1ex}{$15$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\\ \mathrm{n}\in \mathrm{\mathbb{N}}\end{array}\right.\iff \mathrm{n} = \varnothing \end{gathered}$$

$$\begin{gathered} Ответ: а) \{\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}; -\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}; -\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+2\mathrm{\pi n}, \mathrm{n} \in \mathrm{\mathbb{N}}\left\} \\ б\right) \{\raisebox{1ex}{$7\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4; $}\right.2\mathrm{\pi }; 3\mathrm{\pi }\} \end{gathered}$$