Giải phương trình: 3sin3x−√3cos9x=1+4sin^33x

Hướng dẫn giải:

Ta có: (*) \( \Leftrightarrow \left( 3\sin 3x-4{{\sin }^{3}}3x \right)-\sqrt{3}\cos 9x=1\Leftrightarrow \sin 9x-\sqrt{3}\cos 9x=1 \)

 \( \Leftrightarrow \frac{1}{2}\sin 9x-\frac{\sqrt{3}}{2}\cos 9x=\frac{1}{2}\Leftrightarrow \sin \left( 9x-\frac{\pi }{3} \right)=\frac{1}{2}=\sin \frac{\pi }{6} \)

 \( \Leftrightarrow \left[ \begin{align}  & 9x-\frac{\pi }{3}=\frac{\pi }{6}+k2\pi  \\  & 9x-\frac{\pi }{3}=\frac{5\pi }{6}+k2\pi  \\ \end{align} \right. \) \( \Leftrightarrow \left[ \begin{align}  & x=\frac{\pi }{18}+\frac{k2\pi }{9} \\  & x=\frac{7\pi }{54}+\frac{k2\pi }{9} \\ \end{align} \right.,\text{ }k\in \mathbb{Z} \).