7.
\(\sqrt{3}cos3x+sin3x=\sqrt{2}\)
\(\Leftrightarrow\dfrac{\sqrt{3}}{2}cos3x+\dfrac{1}{2}sin3x=\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\dfrac{\pi}{3}cos3x+cos\dfrac{\pi}{3}sin3x=\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\left(3x+\dfrac{\pi}{3}\right)=sin\dfrac{\pi}{4}\)
\(\Leftrightarrow\left[{}\begin{matrix}3x+\dfrac{\pi}{3}=\dfrac{\pi}{4}+k2\pi\\3x+\dfrac{\pi}{3}=\pi-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{36}+\dfrac{k2\pi}{3}\\x=\dfrac{5\pi}{36}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
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