Задача 20. Решите уравнение: \(3{\sin ^2}3x = {\cos ^2}3x\)
Ответ
ОТВЕТ: \( \pm \dfrac{\pi }{{18}} + \dfrac{{\pi k}}{3};\,\,\,\,\,\,\,k \in Z.\)
Решение
\(3{\sin ^2}3x = {\cos ^2}3x.\)
Однородное тригонометрическое уравнение второй степени:
\(3{\sin ^2}3x = {\cos ^2}3x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,3{\rm{t}}{{\rm{g}}^2}3x = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}3x = \dfrac{1}{3}\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{\rm{tg}}3x = \dfrac{1}{{\sqrt 3 }},\,\,}\\{{\rm{tg}}3x = -\dfrac{1}{{\sqrt 3 }}}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{3x = \dfrac{\pi }{6} + \pi k,\,}\\{3x = -\dfrac{\pi }{6} + \pi k}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{\pi }{{18}} + \dfrac{{\pi k}}{3},\,\,\,\,}\\{x = -\dfrac{\pi }{{18}} + \dfrac{{\pi k}}{3},}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \pm \dfrac{\pi }{{18}} + \dfrac{{\pi k}}{3},\,\,\,\,k\, \in \,Z\)
Ответ: \( \pm \dfrac{\pi }{{18}} + \dfrac{{\pi k}}{3};\,\,\,\,\,\,\,k \in Z.\)