  We examine implications of angles having their own dimension, in the same
sense as do lengths, masses, {\it etc.} The conventional practice in scientific
applications involving trigonometric or exponential functions of angles is to
assume that the argument is the numerical part of the angle when expressed in
units of radians. It is also assumed that the functions are the corresponding
radian-based versions. These (usually unstated) assumptions generally allow one
to treat angles as if they had no dimension and no units, an approach that
sometimes leads to serious difficulties. Here we consider arbitrary units for
angles and the corresponding generalizations of the trigonometric and
exponential functions. Such generalizations make the functions complete, that
is, independent of any particular choice of unit for angles. They also provide
a consistent framework for including angle units in computer algebra programs.
