3.
\(sin2x-sin2x.cosx=0\)
\(\Leftrightarrow sin2x\left(1-cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\cosx=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{2}\\x=k2\pi\end{matrix}\right.\)
\(\Leftrightarrow x=\dfrac{k\pi}{2}\)
4.
\(sin3x-cos2x=0\)
\(\Leftrightarrow sin3x=cos2x\)
\(\Leftrightarrow cos\left(\dfrac{\pi}{2}-3x\right)=cos2x\)
\(\Leftrightarrow\dfrac{\pi}{2}-3x=\pm2x+k2\pi\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{\pi}{2}-3x=2x+k2\pi\\\dfrac{\pi}{2}-3x=-2x+k2\pi\end{matrix}\right.\)
...
5.
\(sin\dfrac{x}{2}.cos\dfrac{\pi}{3}+sin\dfrac{\pi}{3}.cos\dfrac{x}{2}=\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(\dfrac{x}{2}+\dfrac{\pi}{3}\right)=\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}+\dfrac{\pi}{3}=\dfrac{\pi}{6}+k2\pi\\\dfrac{x}{2}+\dfrac{\pi}{3}=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{3}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\)
