Question

Giải phương trình: √3.cos2x-sin2x = √3.sinx + cosx

Answer 1

\(\sqrt{3}cos2x-sin2x=\sqrt{3}sinx+cosx\)

\(\Leftrightarrow\sqrt{3}cos2x-\sqrt{3}sinx-sin2x-cosx=0\)

\(\Leftrightarrow\sqrt{3}\left(1-2sin^2x-sinx\right)-2sinx.cosx-cosx=0\)

\(\Leftrightarrow-\sqrt{3}\left(sinx+1\right)\left(2sinx-1\right)-cosx\left(2sinx-1\right)=0\)

\(\Leftrightarrow\left(2sinx-1\right)\left[\sqrt{3}\left(sinx+1\right)+cosx\right]=0\)

\(\Leftrightarrow\left(2sinx-1\right)\left(\sqrt{3}sinx+cosx+\sqrt{3}\right)=0\)

\(\Leftrightarrow sinx=\dfrac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k2\pi\\x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)