The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pöschl—Teller (tPT) potential including a Coulomb-like tensor interaction with arbitrary spin—orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ ± 1)r⁻². In view of spin and pseudo-spin (p-spin) symmetries, the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method (AIM). We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ. The non-relativistic limit is also obtained.