11.
\(sin2x-cos3x=0\)
\(\Leftrightarrow cos\left(\dfrac{\pi}{2}-2x\right)-cos3x=0\)
\(\Leftrightarrow-2sin\left(\dfrac{\pi}{4}+\dfrac{x}{2}\right).sin\left(\dfrac{\pi}{4}-\dfrac{5x}{2}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(\dfrac{\pi}{4}+\dfrac{x}{2}\right)=0\\sin\left(\dfrac{\pi}{4}-\dfrac{5x}{2}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{\pi}{4}+\dfrac{x}{2}=k\pi\\\dfrac{\pi}{4}-\dfrac{5x}{2}=k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{2}+k2\pi\\x=\dfrac{\pi}{10}-\dfrac{k2\pi}{5}\end{matrix}\right.\)
12.
\(4sinx.cosx.cos2x=1\)
\(\Leftrightarrow2sin2x.cos2x=1\)
\(\Leftrightarrow sin4x=1\)
\(\Leftrightarrow4x=\dfrac{\pi}{2}+k2\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{8}+\dfrac{k\pi}{2}\)
14.
\(sinx.cos3x=sin2x.cos4x\)
\(\Leftrightarrow\dfrac{1}{2}\left(sin4x-sin2x\right)=\dfrac{1}{2}\left(sin6x-sin2x\right)\)
\(\Leftrightarrow sin4x=sin6x\)
\(\Leftrightarrow\left[{}\begin{matrix}4x=6x+k2\pi\\4x=\pi-6x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-k\pi\\x=\dfrac{\pi}{10}+\dfrac{k\pi}{5}\end{matrix}\right.\)
13.
\(16cosx.cos2x.cos4x.cos8x=sin3x\)
\(\Leftrightarrow16sinx.cosx.cos2x.cos4x.cos8x=sin3x.sinx\)
\(\Leftrightarrow8sin2x.cos2x.cos4x.cos8x=sin3x.sinx\)
\(\Leftrightarrow4sin4x.cos4x.cos8x=sin3x.sinx\)
\(\Leftrightarrow2sin8x.cos8x=sin3x.sinx\)
\(\Leftrightarrow sin16x=sin3x.sinx\)
Đề lỗi không nhỉ.
