How is the phase of the applied voltage related to the phase of the induced emf in an inductor?
A sinusoidal AC voltage will produce a sinusoidal-shaped current in the inductor. However, that current lags the voltage by 90 degrees because: -

\$v = L\dfrac{di}{dt}\$ or 

\$i = \dfrac{1}{L} \int v\cdot dt\$ and this produces the well-known 90 degrees lag seen below: -



Current flow produces magnetic flux proportional to that current and, a changing magnetic flux produces a back-emf in accordance with to Faraday's Law of induction: -



So, from the originating sinusoidal voltage, you get a current that is 90 degrees lagging and this produces a flux whose rate of change is in phase with the original voltage hence, the back emf is 180 degrees out of phase to the original voltage due to the "minus" sign in Faraday's law.