Capacitor, independent voltage source, phase shift, ' EXO N°4 Vidéo 1, 1'

A capacitor with capacitance "C" is subjected to a sinusoidal voltage. Determine the expression for the current passing through the capacitor and the variation of the current intensity over time, highlighting the phase shift between the voltage and the current.
Note: When analyzing a circuit with alternating voltage or current, calculations are not performed in the time domain because adding and multiplying trigonometric functions such as cos(W.t) and sin(W.t) in the time domain, which depend on the variable 't', are complicated, lengthy, and tedious. Therefore, to simplify the analysis, cos(W.t) and sin(W.t) are transformed into complex numbers expressed in terms of magnitude and phase, known as phasors. Similarly, inductances L and capacitance C are transformed into complex numbers, which can be expressed in terms of magnitude and phase, i.e., in the polar form of a complex number, depending on the operation to be performed (addition, subtraction, multiplication, or division).
For a circuit with only 3 or 4 reactive elements L or C, a time-domain analysis based on the variable 't' quickly becomes unproductive and thus impractical. This is the sole reason why magnitudes and phases are used. Moreover, developing formulas for three-phase systems and especially symmetrical components using trigonometric functions is inextricable, if not impossible. One must use magnitudes and phases, as these are essentially vectors represented in polar form that can be added vectorially.
The method involves converting L and C into reactances expressed as complex numbers using the imaginary unit j, and possibly expressing them in terms of magnitude and phase. While sin(W.t) is converted to cos(W.t), then cos(W.t) is transformed into magnitude and phase (or phasor). Adding or multiplying magnitudes with magnitudes and phases with phases is extremely simple. The magnitudes and phases of voltages and currents also easily visualize their phase shifts.
Important note: When analyzing a circuit using magnitudes and phases, or simply using complex numbers, it means the circuit is analyzed in the frequency domain, and the results found belong to the frequency domain since they do not contain the variable 't'.
One can transform the magnitude and phase of each voltage and current found in the frequency domain back into their equivalent expressions in the time domain depending on the variable 't'.
It is sufficient to write the cosine function for each voltage and current, as phasors, i.e., magnitudes and phases, are derived from cosines:
V(t) = COS(W.t + phaseV°) and I(t) = COS(W.t + phaseI°), where W is known and in [Rad/s], phaseV° represents the phase angle of a voltage, and phaseI° is the phase angle of a current.