Maybe someone else can solve this one by some algebraic means......but here's a graphical solution.........https://www.desmos.com/calculator/bdgkyrfxh1
It looks like the "x's" that make this true are at 78.5°, 233.1° and 306.9° on the interval [0, 360]
5sinx= -4(5cosx-1)/ (5cosx-1)
sinx=-4/5 which is 306.9 degrees.
I have difficulty with the other values (sleepy).