Question

\(\sqrt{2}sin\left(2x-60^0\right)-2sin\left(x-30^0\right)=\sqrt{2}cos\left(2x-60^0\right)\)

Answer 1

Đặt \(x-30^0=a\)

\(\sqrt{2}sin2a-2sina=\sqrt{2}cos2a\)

\(\Leftrightarrow sin2a-cos2a=\sqrt{2}sina\)

\(\Leftrightarrow\sqrt{2}sin\left(2a-45^0\right)=\sqrt{2}sina\)

\(\Leftrightarrow sin\left(2a-45^0\right)=sina\)

\(\Leftrightarrow sin\left(2x-105^0\right)=sin\left(x-30^0\right)\)

\(\Rightarrow\left[{}\begin{matrix}2x-105^0=x-30^0+k360^0\\2x-105^0=180^0-x+30^0+k360^0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=75^0+k.360^0\\x=105^0+k.120^0\end{matrix}\right.\)