$ y = 5\sin x - 12\cos x + 15\\ y = 13\left( {\dfrac{5}{{13}}\sin x - \dfrac{{12}}{{13}}\cos x} \right) + 15\\ y = 13\sin \left( {x - \alpha } \right) + 15\left( {\alpha = \arccos \left( {\dfrac{5}{{13}}} \right)} \right)\\ - 1 \le \sin \left( {x - \alpha } \right) \le 1 \Rightarrow - 13 \le \sin \left( {x - \alpha } \right) \le 13\\ \Rightarrow 2 \le y \le 28\\ \Rightarrow \left\{ \begin{array}{l} \min y = 2\\ \max y = 28 \end{array} \right. $
