Однородные тригонометрические уравнения. Задача 13

Задача 13. Решите уравнение: \({\sin ^2}3x = \sin 3x\cos 3x\)

Ответ

ОТВЕТ: \(\dfrac{{\pi k}}{3};\quad \dfrac{\pi }{{12}} + \dfrac{{\pi k}}{3};\quad \,k \in Z.\)

Решение

\({\sin ^2}3x = \sin 3x\cos 3x\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\sin ^2}3x-\sin 3x\cos 3x = 0\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,\sin 3x\left( {\sin 3x-\cos 3x} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin 3x = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sin 3x-\cos 3x = 0.}\end{array}} \right.\)

\(\sin 3x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,3x = \pi k\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{{\pi k}}{3},\,\,\,\,\,\,\,\,k\, \in \,Z.\)

Уравнение \(\sin 3x-\cos 3x = 0\) является однородным тригонометрическим уравнением первой степени:

\(\sin 3x-\cos 3x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}3x-1 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{tg}}3x = 1\,\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\,\,3x = \dfrac{\pi }{4} + \pi k\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = \dfrac{\pi }{{12}} + \dfrac{{\pi k}}{3},\,\,\,\,\,\,\,\,\,k\, \in \,Z.\)

Ответ: \(\dfrac{{\pi k}}{3};\quad \dfrac{\pi }{{12}} + \dfrac{{\pi k}}{3};\quad \,k \in Z.\)