Задача 3. Решите неравенство: \(\cos \left( {2x-\dfrac{\pi }{4}} \right) > \dfrac{{\sqrt 2 }}{2}.\)
Ответ
ОТВЕТ: \(\left( {\pi k;\,\,\dfrac{\pi }{4} + \pi k} \right),\,\,\,k \in Z.\)
Решение
\(\cos \left( {2x-\dfrac{\pi }{4}} \right) > \dfrac{{\sqrt 2 }}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,-\dfrac{\pi }{4} + 2\pi k < 2x-\dfrac{\pi }{4} < \dfrac{\pi }{4} + 2\pi k\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,2\pi k < 2x < \dfrac{\pi }{2} + 2\pi k\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\pi k < x < \dfrac{\pi }{4} + \pi k,\,\,\,k\, \in \,Z.\)
Ответ: \(\left( {\pi k;\dfrac{\pi }{4} + \pi k} \right),\,\,\,k\, \in \,Z.\)