1.\(sin8x=-\dfrac{1}{2}\Rightarrow sin8x=sin\dfrac{-\Pi}{6}\Rightarrow\left[{}\begin{matrix}8x=\dfrac{-\Pi}{6}+k2\Pi\\8x=\Pi-\left(-\dfrac{\Pi}{6}\right)+k2\Pi\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\dfrac{-\Pi}{48}+\dfrac{k\Pi}{4}\\x=\dfrac{7\Pi}{48}+\dfrac{k\Pi}{4}\end{matrix}\right.\)
2. \(sin9x+cos9x=0\Rightarrow sin9x=-cos9x\Rightarrow\dfrac{sin9x}{cos9x}=\dfrac{-cos9x}{cos9x}\Rightarrow tan9x=-1\)
\(\Rightarrow9x=\dfrac{-\Pi}{4}+k\Pi\Rightarrow x=\dfrac{-\Pi}{36}+\dfrac{k\Pi}{9}\)
3. \(cos5x=sin2x\Rightarrow cos5x=cos\left(\dfrac{\Pi}{2}-2x\right)\Rightarrow\left[{}\begin{matrix}5x=\dfrac{\Pi}{2}-2x+k2\Pi\\5x=-\dfrac{\Pi}{2}+2x+k2\Pi\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}7x=\dfrac{\Pi}{2}+k2\Pi\\3x=\dfrac{-\Pi}{2}+k2\Pi\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=\dfrac{\Pi}{14}+\dfrac{k2\Pi}{7}\\x=\dfrac{-\Pi}{6}+\dfrac{k2\Pi}{3}\end{matrix}\right.\)
