Question

Giải pt : \(\sqrt{3}\)\(\sin x+\cos x\)\(=\sqrt{2}\)

Answer 1

\(\sqrt{3}\sin x+\cos x=\sqrt{2}\Leftrightarrow\frac{\sqrt{3}}{2}\sin x+\frac{1}{2}\cos x=\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow\sin x\cos\frac{\pi}{6}+\cos x\sin\frac{\pi}{6}=\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow\sin\left(x+\frac{\pi}{6}\right)=\sin\frac{\pi}{4}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{6}=\frac{\pi}{4}+k2\pi\\x+\frac{\pi}{6}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{12}+k2\pi\\x=\frac{7\pi}{12}+k2\pi\end{matrix}\right.\left(k\in Z\right)\)