Find the value of `x` in `[-pi,pi]` for which `f(x)=sqrt((log)_2(4sin^2x-2sqrt(3)sinx-2sinx+sqrt(3)+1))` is defined.

Answer

`x in [-pi,(pi)/(6)] cup [(pi)/(3),(2pi)/(3)] cup [(5pi)/(6),pi]`

Explanation

`f(x)=sqrt(log_(2)(4sin^(2)x-2sqrt(3)sinx-2sinx+sqrt(3)+1))` is defined if
`log_(2)(4sin^(2)x-2sqrt(3)sinx-2sinx+sqrt(3)+1) ge 0`
or `4sin^(2)x-2sinx(sqrt(3)+1)+sqrt(3)+1 ge 1`
or ` sin^(2)x-sinx ((sqrt(3))/(2)+(1)/(2)) +(sqrt(3))/(4) ge 0`
or ` (sinx-(sqrt(3))/(2)))(sinx-(1)/(2)) ge 0`
i.e., `-1 le sinx le (1)/(2) " or " (sqrt(3))/(2) le sinx le 1`
or `x in [-pi,(pi)/(6)] cup [(pi)/(3),(2pi)/(3)] cup [(5pi)/(6),pi]`