solve an equation on an interval

sin(x+ pi/6) + sin(x- pi/6 )= sqr(3/2)

lemma:
sin(a)+sin(b) = 2*sin( (a+b)/2 ) * cos( (a-b)/2 )

sin(x+ pi/6) + sin(x- pi/6 )= sqr(3/2)
2*sin( (x+ pi/6 +x- pi/6 )/2 ) * cos( (x+ pi/6 -(x- pi/6) )/2 ) =sqr(3/2)
2*sin( x ) * cos( pi/6 ) = sqr(3/2)
2*sin( x ) * sqrt(3)/2 = sqr(3/2)
2*sin( x ) = (sqr(3/2)) / (sqrt(3)/2)
2*sin( x ) = sqrt(2)
sin(x) = sqrt(2)/2

EdDad28:

Solve the equation on the interval [0,2pi)

x = acos( sqrt(2)/2 ) = pi/4