  General relativistic effects are strong near the black hole of an X-ray
binary and significantly impact the total energy released at the innermost
accretion disk's region. Our goal is to fully incorporate the black hole's spin
and all the general relativistic effects on the observed spectra coming from
X-ray binary systems while maintaining the simplicity of the standard disk
model. That is possible by appropriately shifting only the disk's inner radius.
We employ some of the most efficient pseudo-Newtonian potentials around Kerr
black holes and derive two generalized disk temperature profiles, thus
incorporating the spin's contribution to the thermal spectra. Then, we
associate the observed radiative efficiency with the emission pattern featuring
all the relativistic effects included in the kerrbb model, obtaining an
expression about the modified inner radius of the disk. Moreover, we apply this
method to Cygnus X-1 by fitting the observational data obtained during its
high/soft and hard/low spectral states. The fully relativistic spectra are
reproduced to a very good approximation with an error margin of 0.03-4%. The
disk is parameterized by a modified innermost radius within the range of
$(0.2-2)R_{ISCO}$, depending on the source's viewing angle and black hole spin.
Relativistic effects near the black hole make an otherwise standard accretion
disk with inclination $\theta <60^{\circ}$ seem truncated to larger radii to a
distant observer. On the other hand, an edge-on view of the disk gives the
perspective of being pulled closer to the central object than the respective
ISCO radius. In addition, we show that the observational data of Cygnus X-1 can
be satisfactorily fitted by employing a reasonably simple lepto-hadronic jet
model and a hybrid thermal/non-thermal corona along with the Kerr-adjusted
standard accretion disk.
