Giải phương trình: sinx+cosx+1+sin2x+cos2x=0

Hướng dẫn giải:

Ta có: (*)\(\Leftrightarrow \sin x+\cos x+2\sin x\cos x+2{{\cos }^{2}}x=0\)

\(\Leftrightarrow \sin x+\cos x+2\cos x(\sin x+\cos x)=0\Leftrightarrow (\sin x+\cos x)(1+2\cos x)=0\)

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