Question

2 sin x + 2 cos x -\(\sqrt{2}\)=0

Answer 1

\(\Leftrightarrow2\sin x+2\cos x=\sqrt{2}\)

\(\Leftrightarrow\sin x\cdot\dfrac{1}{\sqrt{2}}+\cos x\cdot\dfrac{1}{\sqrt{2}}=\dfrac{1}{2}\)

\(\Leftrightarrow\sin\left(x+\dfrac{\Pi}{4}\right)=\dfrac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\Pi}{4}=\dfrac{\Pi}{6}+k2\Pi\\x+\dfrac{\Pi}{4}=\dfrac{5}{6}\Pi+k2\Pi\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{1}{12}\Pi+k2\Pi\\x=\dfrac{7}{12}\Pi+k2\Pi\end{matrix}\right.\)