RC circuit with sinusoidal voltage - initial behavior?
I believe you answered your own question. 
The technical term to describe the response of the system after the transients "die out" would be "steady state". 

You have implicitly set the initial condition:
$$
q(0) = 0
$$
As Felthry pointed out in his comment, if you were to calculate the initial condition that produced the right "starting point" so that the sine wave "lined up" properly (in phase and amplitude) at $$t=0$$  then you wouldn't see the transient.  For instance, if you are able to determine the steady state response:
$$
q_\text{steady state}(t) = A\cos(\omega t + \phi),
$$
by measuring the amplitude and phase,
then you just set the initial condition as follows:
$$
q(0) = q_\text{steady state}(0) = A\cos\phi
$$
and you wouldn't see any transient.