Question

\(2\sin\left(\frac{\pi}{4}+x\right)+\sin\left(x-\frac{\pi}{4}\right)=\frac{3\sqrt{2}}{2}\)

Answer 1

\(\Leftrightarrow2\sqrt{2}sin\left(x+\frac{\pi}{4}\right)+\sqrt{2}sin\left(x-\frac{\pi}{4}\right)=3\)

\(\Leftrightarrow2\left(sinx+cosx\right)+sinx-cosx=3\)

\(\Leftrightarrow3sinx+cosx=3\)

\(\Leftrightarrow\frac{3}{\sqrt{10}}sinx+\frac{1}{\sqrt{10}}cosx=\frac{3}{\sqrt{10}}\)

Đặt \(cosa=\frac{3}{\sqrt{10}}\) với \(a\in\left(0;\pi\right)\)

\(\Leftrightarrow sinx.cosa+cosx.sina=cosa\)

\(\Leftrightarrow sin\left(x+a\right)=sin\left(\frac{\pi}{2}-a\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}x+a=\frac{\pi}{2}-a+k2\pi\\x+a=a+\frac{\pi}{2}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}-2a+k2\pi\\x=\frac{\pi}{2}+k2\pi\end{matrix}\right.\)