Воспользуемся свойством пропорции:
\(\dfrac{{4\sin \alpha + 5\cos \alpha + 4}}{{2\sin \alpha -3\cos \alpha -8}} = \dfrac{{-1}}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,2\left( {4\sin \alpha + 5\cos \alpha + 4} \right) = -\left( {2\sin \alpha -3\cos \alpha -8} \right)\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,8\sin \alpha + 10\cos \alpha + 8 = -2\sin \alpha + 3\cos \alpha + 8\,\,\,\,\, \Leftrightarrow \,\,\,\,\,10\sin \alpha = -7\cos \alpha \,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\dfrac{{\sin \alpha }}{{\cos \alpha }} = \dfrac{{-7}}{{10}}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{\rm{tg}}\alpha = -0,7.\)
Ответ: \(-0,7.\)