Воспользуемся свойством пропорции:
\(\dfrac{{3\sin \alpha -2\cos \alpha -2}}{{5\sin \alpha -2\cos \alpha + 8}} = \dfrac{{-1}}{4}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,4\left( {3\sin \alpha -2\cos \alpha -2} \right) = -\left( {5\sin \alpha -2\cos \alpha + 8} \right)\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,12\sin \alpha -8\cos \alpha -8 = -5\sin \alpha + 2\cos \alpha -8\,\,\,\,\, \Leftrightarrow \,\,\,\,\,10\cos \alpha = 17\sin\alpha \,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\dfrac{{\cos \alpha }}{{\sin\alpha }} = \dfrac{{17}}{{10}}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{\rm{ctg}}\alpha = 1,7.\)
Ответ: 1,7.