  Ba$_3$CoSb$_2$O$_9$ approximates the two-dimensional spin-1/2
triangular-lattice Heisenberg antiferromagnet. This compound displays
magnetic-field-induced quantum phase transitions, including the
1/3-magnetization-plateau, but its magnetization processes for the magnetic
field $H$ parallel and perpendicular to the $c$ axis are different due to the
weak easy-plane anisotropy and the weak interlayer antiferromagnetic exchange
interaction. To elucidate how the quantum phase transitions change between
these two field directions, we measured the field-angle dependence of the
magnetization process in Ba$_3$CoSb$_2$O$_9$ using pulsed high magnetic fields.
We compared obtained magnetic field-field angle phase diagram with those
obtained by the large-size cluster mean-field method combined with a scaling
scheme and the semiclassical theory. We also found a narrow 1/3-magnetization
plateau and a high-field transition with a small magnetization jump for
$H\,{\parallel}\,c$, not observed in the previous studies.
