$\begin{array}{l}
y = \cos 2x - \sin x + 3\\
y = 1 - 2{\sin ^2}x - \sin x + 3\\
y = - 2{\sin ^2}x - \sin x + 4\\
t = \sin x\left( { - 1 \le t \le 1} \right)\\
\Rightarrow y = f\left( t \right) = - 2{t^2} - t + 4\\
\Rightarrow t = - \dfrac{b}{{2a}} = - \dfrac{1}{4}\\
\Rightarrow f\left( { - \dfrac{1}{4}} \right) = \dfrac{{33}}{8} \Rightarrow \max y = \dfrac{{33}}{8}\\
f\left( { - 1} \right) = 3,f\left( 1 \right) = 1 \Rightarrow \min y = 1\\
\left\{ \begin{array}{l}
\min y = 1 \Rightarrow \sin x = 1\\
\max y = \dfrac{{33}}{8} \Rightarrow \sin x = - \dfrac{1}{4}
\end{array} \right.
\end{array}$
