Vrms × Irms = Apparent Power
Your equation here is actually the equation for the magnitude of your real power (P) or true power as you call it, not apparent power (S).
However, if your reactive power is equal to zero VARs, your equation would produce the correct value. This is because when a load is purely resistive (only real power) the apparent power and real power are equal.
Keep in mind that apparent power is the magnitude of some number of Volt-Amperes. Complex power (S) is the same value with the phase angle. This is a subtle nuance and most people interchange the apparent power and complex power while calling both the apparent power. I will try to distinguish to make it easier for you to understand.
Apparent Power² = Real Power² + Reactive Power²
(Apparent Power = Sq. Root of Real Power² + Reactive Power²)
Complex Power = Real Power + Reactive Power
(S∠θ = P∠θ + jQ∠θ)
Power Factor for the calculation of True Power
The power factor can be calculated by the taking the cosine of the phase angle difference between your voltage and your current.
Power Factor = cos(voltageθ - currentθ)
There are other ways of calculating the power factor by using the complex power, apparent power, real power and reactive power.
It would seem the easiest to me to calculate your real power, then determine your impedance to get your reactive power, then calculate your power factor=cos(arctan(Q/P). But if you can't can't determine your impedance, then that means you can't calculate your phase quantities for complex or reactive power. The power factor is about calculating an angle so if you don't have the complex phase quantity which includes an angle, then you can't use that route. It is really important to try to work in the phasor domain as opposed to the time domain when working with power.
True Power (Vrms × Irms × Power Factor = True Power)
Your equation here actually shows you are calculating:
True Power = True Power * Power Factor
By rearranging the terms you are saying your Power Factor is equal to 1. This may be true but I do not think this is what you are trying to calculate.