Воспользуемся свойством пропорции:
\(\frac{{3\sin \alpha — 5\cos \alpha + 2}}{{\sin \alpha + 3\cos \alpha + 6}} = \frac{1}{3}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,3\left( {3\sin \alpha — 5\cos \alpha + 2} \right) = \sin \alpha + 3\cos \alpha + 6\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,9\sin \alpha — 15\cos \alpha + 6 = \sin \alpha + 3\cos \alpha + 6\,\,\,\,\, \Leftrightarrow \,\,\,\,\,8\sin \alpha = 18\cos \alpha \,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{18}}{8}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,tg\alpha = 2,25.\)
Ответ: 2,25.