Question

giải phương trình: \(\sqrt{3}sinx+cosx=\sqrt{2}\)

Answer 1

\(\Leftrightarrow\frac{\sqrt{3}}{2}sinx+\frac{1}{2}cosx=\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow sinx.cos\frac{\pi}{6}+cosx.sin\frac{\pi}{6}=sin\frac{\pi}{4}\)

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

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

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