$$\eqalign{ & 9)\,\,y = {1 \over 2}\sin x + {{\sqrt 3 } \over 2}\cos x \cr & y = \sin x\cos {\pi \over 3} + \cos x\sin {\pi \over 3} \cr & y = \sin \left( {x + {\pi \over 3}} \right) \cr & \Rightarrow - 1 \le y \le 1 \cr & \Rightarrow \min y = - 1 \Leftrightarrow \sin \left( {x + {\pi \over 3}} \right) = - 1 \cr & \Leftrightarrow x + {\pi \over 3} = - {\pi \over 2} + k2\pi \Leftrightarrow x = - {{5\pi } \over 6} + k2\pi \cr & \max y = 1 \Leftrightarrow \sin \left( {x + {\pi \over 3}} \right) = 1 \cr & \Leftrightarrow x + {\pi \over 3} = {\pi \over 2} + k2\pi \Leftrightarrow x = {\pi \over 6} + k2\pi \cr & 12)\,y = \sqrt {{1 \over 2} - \sin x} \cr & - 1 \le \sin x \le 1 \cr & \Leftrightarrow 1 \ge - \sin x \ge - 1 \cr & \Leftrightarrow {3 \over 2} \ge {1 \over 2} - \sin x \ge - {1 \over 2} \cr & \Leftrightarrow \sqrt {{3 \over 2}} \ge \sqrt {{1 \over 2} - \sin x} \ge 0 \cr & \Rightarrow \min y = 0 \Leftrightarrow \sin x = {1 \over 2} \Leftrightarrow \left[ \matrix{ x = {\pi \over 6} + k2\pi \hfill \cr x = {{5\pi } \over 6} + k2\pi \hfill \cr} \right. \cr & \max y = \sqrt {{3 \over 2}} \Leftrightarrow \sin x = - 1 \Leftrightarrow x = - {\pi \over 2} + k2\pi \cr} $$

\[\begin{array}{l} 9)\,\,y = \frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x = \sin \left( {x + \frac{\pi }{3}} \right)\\ Ta\,\,co:\,\, - 1 \le \sin \left( {x + \frac{\pi }{3}} \right) \le 1\\ \Rightarrow Min\,\,y = - 1 \Leftrightarrow \sin \left( {x + \frac{\pi }{3}} \right) = - 1\\ Maxy = 1 \Leftrightarrow \sin \left( {x + \frac{\pi }{3}} \right) = 1.\\ 12)\,\,y = \sqrt {\frac{1}{2} - \sin x} \\ Ta\,\,co:\,\,\, - 1 \le \sin x \le 1\\ \Rightarrow - 1 \le - \sin x \le 1\\ \Leftrightarrow - \frac{1}{2} \le \frac{1}{2} - \sin x \le \frac{3}{2}\\ \Rightarrow 0 \le \sqrt {\frac{1}{2} - \sin x} \le \frac{{\sqrt 6 }}{2}\\ \Rightarrow Min\,y = 0 \Leftrightarrow \sin x = \frac{1}{2}\\ Maxy = \frac{{\sqrt 6 }}{3} \Leftrightarrow \sin x = - 1. \end{array}\]