Voltage Across a Resistor RLC

Given an RLC circuit I want to model the in- and out-of-phase of the voltage across the resistor $R$ as a function of the frequency $omega$ of the AC-supplied voltage $U = U_0 cos(omega)$. The RLC components are connected in series, and the inductive coil of inductance $L$ has a resistor, $R_{coil}$, connected to it as well (thus also in series with $R$).

So far, what I have is as follows:

Using Kirchoff’s laws:

$$ U = I(omega)R_{sigma} + L frac{d I(omega)}{dt} + frac{Q}{C}$$

and the equation for the frequency dependent current draw can be written as:

$$ I(omega) = frac{U_0}{Z(omega)} = frac{U_0}{R_{sigma} + ileft( omega L – frac{1}{omega C}right)}$$

where $R_{sigma} = R + R_{coil}$, $U_0$ and $C$ the capacitance of the capacitor.

The voltage across the resistor is then
$$ V_R = I(omega) R $$

If I understand correctly, are the in- and out-of-phase components of the voltage $V_R$ simply the real (in-phase) and imaginary (out-of-phase) parts of the above equation? i.e.,
$$V_{R, real} = |I(omega)|R = frac{U_0R}{sqrt{R_{sigma}^2 +left( omega L – frac{1}{omega C}right)^2}}$$

What would the imaginary component look like?

A follow up question would be that if I was to measure the in- and out-of-phase components of $V_R$ using a lock-in-amplifier, how would I use the equations above to fit the theoretical curves derived above to the measured data, if I did not have values of $C, L, R, R_{coil}$?

I think it might have to do with finding the resonance frequency $omega_0 = frac{1}{sqrt{LC}}$ using the measured values of $V_R$ but I am not sure how this is possible.