Задача 2. Решите уравнение: \(\cos x\left( {2\cos x + \sqrt 2 } \right) = 0\)
Ответ
ОТВЕТ: \(\dfrac{\pi }{2} + \pi k;\,\,\,\, \pm \,\dfrac{{3\pi }}{4} + 2\pi k;\quad \,k \in Z.\)
Решение
\(\cos x\left( {2\cos x + \sqrt 2 } \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2\cos x + \sqrt 2 = 0}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,}\\{\cos x = -\dfrac{{\sqrt 2 }}{2}}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow } \right.} \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{\pi }{2} + \pi k,\,\,\,\,\,\,\,\,\,\,\,}\\{x = \pm \dfrac{{3\pi }}{4} + 2\pi k,}\end{array}\,\,\,\,\,\,\,\,k\, \in \,Z.} \right.\)
Ответ: \(\dfrac{\pi }{2} + \pi k;\,\,\,\, \pm \,\dfrac{{3\pi }}{4} + 2\pi k;\quad \,k \in Z.\)