\[\sin{34{^\circ}}\cos{11{^\circ}} =\]
\[= \frac{1}{2} \cdot \left( \sin{45{^\circ}} + \sin{23{^\circ}} \right) =\]
\[= \frac{1}{2} \cdot \left( \frac{\sqrt{2}}{2} + \sin{23{^\circ}} \right) =\]
\[= \frac{\sqrt{2}}{4} + \frac{1}{2}\sin{23{^\circ}}\]