Задача 13. Решите уравнение: \({\sin ^2}x = \dfrac{1}{2}\)
ОТВЕТ: \(\dfrac{\pi }{4} + \dfrac{{\pi k}}{2};\quad k \in Z.\)
\({\sin ^2}x = \dfrac{1}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin x = \dfrac{{\sqrt 2 }}{2},}\\{\sin x = -\dfrac{{\sqrt 2 }}{2}}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{\pi }{4} + 2\pi k,\,\,\,\,\,\,\,}\\{x = \dfrac{{3\pi }}{4} + 2\pi k,\,\,\,\,}\\{x = -\dfrac{\pi }{4} + 2\pi k,\,\,\,}\\{x = -\dfrac{{3\pi }}{4} + 2\pi k\,}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,x = \dfrac{\pi }{4} + \dfrac{{\pi k}}{2},\,\,\,\,\,k\, \in \,Z.\) Ответ: \(\dfrac{\pi }{4} + \dfrac{{\pi k}}{2};\quad k \in Z.\)