27.
\(\cos\left(\frac{\pi}{6}-2x\right)=\sin x\)
\(\Leftrightarrow\sin\left(\frac{\pi}{2}-\frac{\pi}{6}+2x\right)=\sin x\)
\(\Leftrightarrow\sin\left(\frac{\pi}{3}+2x\right)=\sin x\)
\(\Leftrightarrow\frac{\pi}{3}+2x=\pi-x+k2\pi\Leftrightarrow x=\frac{2}{9}\pi+\frac{2}{3}k\pi\)
\(\frac{\pi}{2}< \frac{2}{9}\pi+\frac{2}{3}k\pi< \pi\Leftrightarrow\frac{5}{18}\pi< \frac{2}{3}k\pi< \frac{7}{9}\pi\)
\(\Leftrightarrow\frac{5}{12}< k< \frac{7}{6}\Rightarrow k=1\)
Vậy phương trình có 1 nghiệm thuộc khoảng \(\left(\frac{\pi}{2};\pi\right)\)
19. \(\sin3x=\sin x\Leftrightarrow3x=\pi-x+k2\pi\Rightarrow x=\frac{\pi}{4}+\frac{1}{2}k\pi\)
33. \(DKXD:\left\{{}\begin{matrix}\cos3x\ne0\\\sin2x\ne0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x\ne\frac{\pi}{2}+k\pi\\2x\ne\pi+k\pi\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{6}+\frac{1}{3}k\pi\\x\ne\frac{\pi}{2}+\frac{1}{2}k\pi\end{matrix}\right.\)
\(\tan3x.\cot2x=1\Leftrightarrow\tan3x=\frac{1}{\cot2x}=\tan2x\)
\(\Leftrightarrow3x=\pi+2x+k\pi\Leftrightarrow x=\pi+k\pi\) (t/m)
