Outline: 
Sinusoidal Sources 
Review of Complex Numbers 
AC Steady State Analysis 
Definition of a Phasor 
AC Analysis using Phasors 
This AC Circuit is in Steady State. This means that all of the AC voltages and currents are constant (not flat like DC but unchanging). Note there are many detailed steps, but if you can follow this problem then you will be able to grasp what a phasor is in the next section. 

Note that in this example the e^{j2t} cancelled out of the math. Therefore to determine B we did not need the e^{j2t}. The radian frequency (the '2' in this case) is only needed when differentiating the KVL equation. (see step 3) In summary, the radian frequency determines the coefficients of the equation but after step 3 the radian frequency is simply in the way. 
The phasor for V is: V = 30/45 or V = 30/135 

The phasor for I is: I = 5/30 or I = 5/150 

Here we will convert from sine to cosine: As seen in section one, we subtract 90^{o} to convert from sine to cosine. I = 10 cos(3t 30^{o}  90^{o}) = 10 cos(3t  120^{o}) Also recall from math that cos(wt) = cos(wt +/ 180^{o}) So I = 10 cos(3t  120^{o} + 180^{o}) = 10 cos(3t + 60^{o}) The phasors for I are: I = 10/120 or I = 10/60 

Again we convert from sine to cosine: V = 30 cos(10t  180^{o}  90^{o}) = 30 cos(10t  270^{o}) V = 30 cos(10t  270^{o} + 360^{o}) = 30 cos(10t + 90^{o}) The phasors for V are: V = 30/270 or V = 30/90 

Note that there are many ways of writing one phasor. As long as each phasor value represents the same vector when graphed, then all are equivalent. 
One can see that Ohm's Law, V=IR, holds true for phasors. The above VoltageCurrent relationships for the Inductor, the Capacitor, and the resistor will significantly simplify AC analysis. 