That 90º lag comes from the derivative with respect to time in the Faraday's law of induction:
(Differential form)
(Integral form)
The left hand side of this last equation is the EMF. If those equations didn't have a derivate with respect to time, |E(t)| would be maximum when |B(t)| is maximum, but the (negative) derivative causes a sin(t) in E(t) to turn into a -cos(t), and that is where the 90º lag comes from.