How did you get Sin x = 17.4 / 97.2 ?
Talking about the angles that are not multiples of the angles of 0, 30, 45, 60, 90 grades the exact value - extracting roots - is not so important. In this case, if I don't misinterpret yours written trigonometric equation, the root is
x = (-1)^n arcsin (0.18) + pi n solve(sin (x) = 0.18,x)
+259 
