how to solve sin^-1(cos(2pi/3))?

sin^-1(cos(2pi/3)

cos 2pi/3  = -1/2   ....so...

sin-1(-1/2) = -pi/6

Remember that the domain for inverse sine is     $$-\pi/2\le \pi/2$$

So only 1st and 4th quadrant answers are relevant.  :)