`sin(x + pi/6) - sin(x - (7pi)/6) = sqrt(3)/2` Find all solutions of the equation in the interval `[0, 2pi).`

Use the formula of difference of sinuses

`sin(a)-sin(b)=2sin((a-b)/2)cos((a+b)/2)`

and obtain

`2sin((2pi)/3)cos(x-pi/2)=sqrt(3)/2,`

or `2*sqrt(3)/2*cos(x-pi/2)=sqrt(3)/2,`

or `cos(x-pi/2)=1/2.`

The general solution is `x-pi/2=+-pi/3+2kpi,` or `x=pi/2+-pi/3+2kpi.`

The solutions on the interval `[0, 2pi)` are `x=(5pi)/6` and `x=pi/6.`

Use the formula of difference of sinuses

`sin(a)-sin(b)=2sin((a-b)/2)cos((a+b)/2)`

and obtain

`2sin((2pi)/3)cos(x-pi/2)=sqrt(3)/2,`

or `2*sqrt(3)/2*cos(x-pi/2)=sqrt(3)/2,`

or `cos(x-pi/2)=1/2.`

The general solution is `x-pi/2=+-pi/3+2kpi,` or `x=pi/2+-pi/3+2kpi.`

The solutions on the interval `[0, 2pi)` are `x=(5pi)/6` and `x=pi/6.`