Relation between torque angle and power factor

What is the relationship between torque angle and power factor for a induction motor? Here is my understanding.

Lets say a stator coil produces a magnetic field and the field induces voltage in a rotor coil such that this rotor coil is in the same plane as the stator. There could be two extreme cases at this point:

1. if the rotor coil is purely inductive, there is a out of phase rotor current (90deg lagging) which produces a rotor magnetic field varying 90 deg in lag to the stators field. This phase shift in time is also translated to spatial 90 deg lag from stator flux.

2. If the rotor is purely resistive, then there is a rotor current in phase to voltage and the rotor field aligns to the stator's field.

In other cases, there is some combination of both above points which results in a rotor flux angle between 0to 90 degree lagging to stator field.

So, if the above logic is correct, and assuming power factor is caused only due to rotor, then

1. Rotor coil Inductive rotor == 0 pf == 90 deg torque angle.
2. Rotor coil Resistive load == 1 pf == 0 deg torque angle.

This is my understanding. I can see my logic could be wrong since 90 deg torque angle( maximum torque) is resulted by 0 pf( no useful work done). Can anyone provide me a better logical relation between pf and torque angle? Since I am learning this on my own, you can redirect me to some link as well if I have missed something basic.