Capt Pit Bull is essentially correct except in one detail - with TCAS Version 7 the relative bearing IS used by an internal TCAS collision avoidance function called the Miss Distance Filter: an estimate of the predicted minimum horizontal separation is made based on a linear projection of the aircraft trajectories. If TCAS diagnoses that this prediction is reliable (i.e. neither aircraft is manoeuvring by turning or changing speed) AND that the minimum separation is large enough (i.e. greater than the TCAS parameter DMOD - 0.2NM at low altitude up to 1.1NM at high altitude) then the generation of an RA can be suppressed.
If the predicition is not reliable or the predicted separation is not large enough then an RA will be generated in the normal way.
So the relative bearing is not used to generate RAs, but it can be used (in favourable circumstances) to suppress unnecessary RAs.