Tìm nghiệm của phương trình  \(\sin x + \sqrt 3 \c...

A

- Hướng dẫn giải

Ta có:\(\sin x + \sqrt 3 \cos x = \sqrt 2 \)

\(\begin{array}{l}
\Leftrightarrow \frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x = \frac{{\sqrt 2 }}{2}\\
\Leftrightarrow \cos \frac{\pi }{3}\sin x + \sin \frac{\pi }{3}\cos x = \frac{{\sqrt 2 }}{2}
\end{array}\)

\( \Leftrightarrow \sin \left( {x + \dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 2 }}{2}\)

\( \Leftrightarrow \sin \left( {x + \dfrac{\pi }{3}} \right) = \sin \dfrac{\pi }{4} \)\( \Leftrightarrow \left[ \begin{array}{l}x + \dfrac{\pi }{3} = \dfrac{\pi }{4} + k2\pi \\x + \dfrac{\pi }{3} = \pi  - \dfrac{\pi }{4} + k2\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}x =  - \dfrac{\pi }{{12}} + k2\pi \\x = \dfrac{{5\pi }}{{12}} + k2\pi \end{array} \right.\,\left( {k \in \mathbb{Z}} \right)\)