Giải phương trình: sinx+sin2x+sin3x=cosx+cos2x+cos3x

Hướng dẫn giải:

Phương trình  \( \Leftrightarrow (\sin x+\sin 3x)+\sin 2x=(\cos x+\cos 3x)+\cos 2x \)

 \( \Leftrightarrow 2\sin 2x\cos x+\sin 2x=2\cos 2x\cos x+\cos 2x\Leftrightarrow \sin 2x(2\cos x+1)=\cos 2x(2\cos x+1) \)

 \( \Leftrightarrow (2\cos x+1)(\sin 2x-\cos 2x)=0\Leftrightarrow \left[ \begin{align} & 2\cos x+1=0 \\  & \sin 2x-\cos 2x=0 \\ \end{align} \right. \)

 \( \Leftrightarrow \left[ \begin{align}  & \cos x=-\frac{1}{2}=\cos \frac{2\pi }{3} \\  & \sin 2x=\cos 2x \\ \end{align} \right. \) \( \Leftrightarrow \left[ \begin{align}  & x=\pm \frac{2\pi }{3}+k2\pi  \\  & \tan 2x=1=\tan \frac{\pi }{4} \\ \end{align} \right. \) \( \Leftrightarrow \left[ \begin{align} & x=\pm \frac{2\pi }{3}+k2\pi  \\  & 2x=\frac{\pi }{4}+k\pi  \\ \end{align} \right. \)

 \( \Leftrightarrow \left[ \begin{align} & x=\pm \frac{2\pi }{3}+k2\pi  \\  & x=\frac{\pi }{8}+\frac{k\pi }{2} \\ \end{align} \right.,\text{ }k\in \mathbb{Z} \).