This paper investigates a Rician complex envelope which is subject to line-of-sight (LOS) shadowing. In particular, exact closed-form expressions are obtained for the joint envelopephase distribution, the distribution of the phase, and that of the quadrature and in-phase signal components. Using the new formulation for the phase distribution, we find that for increasing shadowing severity, the phase becomes progressively more disperse. Interestingly, the phase is shown to be unimodal. This illustrates that the relationship which is known to exist between the envelope of Rician fading which undergoes LOS shadowing and Hoyt fading does not extend to the phase. We also provide two applications of our new results which investigate the average bit error probability and the average symbol error probability for phase-based modulation schemes operating in LOS shadowed fading channels. The results are shown to provide excellent agreement with Monte Carlo simulations.