Possibly Evaluating a Trig Function of an Inverse Trig Function

I'm asked to find (root-3)sinx - cosx = -1 on the interval [0, 2pi). It comes in the middle of some questions which ask to evaluate trig functions of inverse trig functions, like cos(sin^-1(1/4)) so that tells me I'm supposed to do something like this at some point, but I can't see how.

I came up with a solution though somewhat inelegant, and I'm wondering if anyone sees a better solution. Mine went: Move the -1 and -cosx to the other sides of the equation, square both sides, factor sinx, solve each factor for 0, test each solution in the interval, obtain only 0 and 4pi/5.