I have reached a stage in a problem where I have a complex current through an inductor in a parallel LCR circuit with a current source (the circuit is a parallel LC circuit with a resistance R in series as well as a driving voltage V0sinωt, converted into the above via a Norton equivalent). I have obtained IL=V0sinωt/R[1+i(ωL/R)-(ω2LC)] by first finding the common voltage across each component. The problem then says that the circuit is excited at the resonant frequency ω0=1/√LC by a voltage cosω0t. I need to calculate IL(t). This reduces my expression to IL=cosω0t/iω0L. Now I need to get this to be real. I just wrote cosω0t=eiω0t and i=eiπ/2 giving IL=cos(ω0t-π/2)/ω0L after taking the real part. Now I'm not sure if this is correct. Besides that, if it is, I don't quite understand why I would be allowed to do that. Why would I just take the real part at the end. It just doesn't seem mathematiclly rigorous and so if somebody could explain the maths behind the approach I would feel more comfortable. Thanks.