Neglecting certain elements in deriving the current of a series RLC circuit

Question

The current at resonance in a series RLC circuit is described in the bottom derivation, along with it being plugged into the formula for finding the voltage across the inductor in the circuitw:

I have two questions regarding this derivation.

1. Isn't the current at resonance supposed to include the phase shift φ? Why is it left out of the equation? Example: I*sin(ωt+φ)
2. When the equation for the current at resonance is plugged into the equation for finding the voltage across the inductor, why is the trigonometric part of the differentiated current equation neglected in the next step?

Answer 1

1. You are basically right. But the phase shift would not change anything in the equations (except the last one). The derivative of sin() is still cos() and the relative phase relation stays the same, no matter what phase sin() has. You can do the math yourself.
2. cos() is removed in the last equation because cos(t=0) is 1. But if you bring your phase shift in play, then it needs to be considered with cos(phi).