  We develop a fully microscopic theory for the calculations of the
angle-dependent properties of unconventional superconductors under a rotated
magnetic field. We employ the quasiclassical Eilenberger equations, and use a
variation of the Brandt-Pesch-Tewordt (BPT) method to obtain a closed form
solution for the Green's function. The equations are solved self-consistently
for quasi-two-dimensional $d_{x^2-y^2}$ ($d_{xy}$) superconductors with the
field rotated in the basal plane. The solution is used to determine the density
of states and the specific heat. We find that applying the field along the gap
nodes may result in minima or maxima in the angle-dependent specific heat,
depending on the location in the T-H plane. This variation is attributed to the
scattering of the quasiparticles on vortices, which depends on both the field
and the quasiparticle energy, and is beyond the reach of the semiclassical
approximation. We investigate the anisotropy across the T-H phase diagram, and
compare our results with the experiments on heavy fermion CeCoIn$_5$.
