We present an alternative to the standard method of solution of the Dirac equation for a particle in a Coulomb potential, which hews closer to the more familiar solution of the nonrelativistic Schrödinger equation. The bound-state wave functions so obtained are also closer in form to the nonrelativistic Schrödinger wave functions and make the transition to this limit more evident. The equivalence of these and the “standard” solution form of the Dirac wave functions is demonstrated in the Appendix, which, at the same time, shows the advantage of a new trigonometric parametrization in the algebraic manipulation of relativistic quantities.