According to sine theorem AB/sinC=BC/sinA, so BC = (AB/SINC) * Sina = (1/sin (5 π/12)) * sin (π/4).
Sin (5 π/12) = sin ((π/4)+(π/6)) = sin (π/4) cos (π/6)+sin (π/6) cos (π/4) = (equation 2/2) (equation 3
The result is BC= root number 3- 1.