Задача 8. Решите неравенство: \({\text{ctg}}\,\left( {x-\dfrac{\pi }{6}} \right) \leqslant -1.\)
Ответ
ОТВЕТ: \(\left[ {\dfrac{{11\pi }}{{12}} + \pi k;\,\,\dfrac{{7\pi }}{6} + \pi k} \right),\,\,\,k \in Z.\)
Решение
\({\rm{ctg}}\left( {x-\dfrac{\pi }{6}} \right) \le -1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\dfrac{{3\pi }}{4} + \pi k \le x-\dfrac{\pi }{6} < \pi + \pi k\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\dfrac{{3\pi }}{4} + \dfrac{\pi }{6} + \pi k \le x < \pi + \dfrac{\pi }{6} + \pi k\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\dfrac{{11\pi }}{{12}} + \pi k \le x < \dfrac{{7\pi }}{6} + \pi k,\,\,\,k\, \in \,Z.\)
Ответ: \(\left[ {\dfrac{{11\pi }}{{12}} + \pi k;\dfrac{{7\pi }}{6} + \pi k} \right),\,\,\,k\, \in \,Z.\)