I am very familar with vectors, series and parallel tank circuits, XL, Xc, R, Z, all the equations in electronics. Never really applied them to motors.
What I'm not 100% sure of, but going to take a guess an educated guess on what you call unity(powerfactor).
When the phase angle between the vector sum and R is zero, the cos of zero is one and the Power Factor is 1, unity.
So, if you were to find the phase angle then the cosine of that angle transulates to the power factor. example. The phase angle is 45 degrees, the cosine of 45 is .707. Therefore you would say the power factor is .707. As the phase angle approaches 90 degrees, the power factor approaches 0. As phase angle approaches zero, power factor approaches 1.
If you can follow this chatter, good luck.
Yes. I have too much time on my hands.
So, is the cos of phase angle = power factor.
Inverse cosine of power factor = phase angle.
I'm kinda of rusty too. Only been retired 10 years. I pitched all my books.