Your equation has a function of 2x. Therefore, we need to expand the interval that you are solving for, because when you take the arcsin of both sides, you must account for the 2x, not just x. So, we expand our interval from 0<x<2(pi) to 0<2x<4(pi).
When we take the arcsin of both sides, you will need to find all of the solutions that exist in (0,4pi). Then, when you have all solutions for 2x, just divide by two FOR ALL SOLUTIONS, and then you will have your needed solutions (I'd assume that there will be four solutions).