2.
\(\Leftrightarrow\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(x-\dfrac{\pi}{3}\right)=sin\left(\dfrac{\pi}{6}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{3}=\dfrac{\pi}{6}+k2\pi\\x-\dfrac{\pi}{3}=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k2\pi\\x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)
4.
\(\Leftrightarrow\dfrac{1}{2}sin4x+\dfrac{\sqrt{3}}{2}cos4x=\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow cos\left(4x-\dfrac{\pi}{6}\right)=cos\left(\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}4x-\dfrac{\pi}{6}=\dfrac{\pi}{4}+k2\pi\\4x-\dfrac{\pi}{6}=-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{48}+\dfrac{k\pi}{2}\\x=-\dfrac{\pi}{48}+\dfrac{k\pi}{2}\end{matrix}\right.\)
6.
\(\Leftrightarrow\dfrac{3}{5}sinx+\dfrac{4}{5}cosx=1\)
Đặt \(\dfrac{3}{5}=cosa\) với \(a\in\left(0;\dfrac{\pi}{2}\right)\)
\(\Rightarrow sinx.cosa+cosx.sina=1\)
\(\Leftrightarrow sin\left(x+a\right)=1\)
\(\Leftrightarrow x+a=\dfrac{\pi}{2}+k2\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{2}-a+k2\pi\)
8.
\(\Leftrightarrow\dfrac{1}{2}cos7x-\dfrac{\sqrt{3}}{2}sin7x=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow cos\left(7x+\dfrac{\pi}{3}\right)=cos\left(\dfrac{3\pi}{4}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}7x+\dfrac{\pi}{3}=\dfrac{3\pi}{4}+k2\pi\\7x+\dfrac{\pi}{3}=-\dfrac{3\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{84}+\dfrac{k2\pi}{7}\\x=-\dfrac{13\pi}{84}+\dfrac{k2\pi}{7}\end{matrix}\right.\)
