Однородные тригонометрические уравнения. Задача 18

Задача 18. Решите уравнение: \(3{\sin ^2}x = {\cos ^2}x\)

Ответ

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

Решение

\(3{\sin ^2}x = {\cos ^2}x.\)

Однородное тригонометрическое уравнение второй степени:

\(3{\sin ^2}x = {\cos ^2}x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,3{\rm{t}}{{\rm{g}}^2}x = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{t}}{{\rm{g}}^2}x = \dfrac{1}{3}\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{{\rm{tg}}\,x = \dfrac{1}{{\sqrt 3 }},\,\,}\\{{\rm{tg}}\,x = -\dfrac{1}{{\sqrt 3 }}}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{\pi }{6} + \pi k,\,\,\,\,}\\{x = -\dfrac{\pi }{6} + \pi k,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \pm \dfrac{\pi }{6} + \pi k,\,\,\,\,\,\,\,\,\,k\, \in \,Z.\)

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