How does a negative current start in the inductor in ac circuit?
When you apply a sinewave voltage to an inductor, current has no option other than to begin at zero - it cannot suddenly adopt a negative value as would be implied in the steady-state waveforms.

In fact it has a name - it's called inrush current and particularly in motors, inductors and transformers it can cause core saturation because, the current rises above the peak of the normal steady-state value: -



This is why some inductive loads are connected to the AC supply when the voltage waveform is at its maximum peak (negative or positive) because that is when the current would naturally flow through a zero-crossing point.

Without core saturation (or other losses) the current would remain having a positive average DC value. Here's a picture that shows what I mean: -



Picture taken from here.

When voltage is applied at a zero crossing the current will remain having a dc value for some time and this dc offset will ebb away with losses due to R1.