Find all solutions of the equation in the interval [0, 2pi):

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

My work:

sin x cos pi/6 + cos x sin pi/6 - sin x cos pi/6 - cos x sin pi/6 = 1/2

2 sin x cos pi/6 = 1/2

(2 sin x) (sqrt3/2) = 1/2

Now this is the part where I am confused,

2 sin x = (1/2) / (sqrt3/2)

2 sin x = (2) / (2 sqrt3)

2 sin x = 1/sqrt 3

sin x = sqrt3/2

x = pi/3, 2pi/3

Is this right?

Any help is appreciated