Уравнения, сводящиеся к простейшим заменой неизвестного. Задача 4

Задача 4. Решите уравнение: \(2{\cos ^2}x-\sqrt 3 \cos x = 0\)

Ответ

ОТВЕТ: \(\dfrac{\pi }{2} + \pi k;\,\,\,\, \pm \,\dfrac{\pi }{6} + 2\pi k;\quad \,k \in Z.\)

Решение

\(2{\cos ^2}x-\sqrt 3 \cos x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\cos x\left( {2\cos x-\sqrt 3 } \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2\cos x-\sqrt 3 = 0}\end{array}\,\,\,\,\,\,\, \Leftrightarrow } \right.\)

\( \Leftrightarrow \,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos x = 0,\,\,\,\,}\\{\cos x = \dfrac{{\sqrt 3 }}{2}}\end{array}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,} \right.\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{\pi }{2} + \pi k,\,\,\,\,\,\,\,\,}\\{x = \pm \dfrac{\pi }{6} + 2\pi k,}\end{array}\,\,\,\,\,\,\,k\, \in \,Z.} \right.\)

Ответ: \(\dfrac{\pi }{2} + \pi k;\,\,\,\, \pm \,\dfrac{\pi }{6} + 2\pi k;\quad \,k \in Z.\)