An introduction to mathematics, by A. N. Whitehead.

184 INTRODUCTION TO MATHEMATICS
to indicate the special function of "sine,"
and "cos" for "f" when we want to indicate
the special function of "cosine." Thus, with
the above meanings for u, v, w, we get
v=sin u, and w =cos u,
where the brackets surrounding the x in f(x)
are omitted for the special functions. The
meaning of these functions sin and cos as
correlating the pairs of numbers u and v, and
u and w is, that the functional relations are to
be found by constructing (cf. fig. 26) an angle
AOP, whose measure "AP divided by OP"
is equal to u, and that then v is the number
given by "PM divided by OP" and w is the
number given by "OM divided by OP."
It is evident that without some further definitions we shall get into difficulties when the
number u is taken too large. For then the arc
AP may be greater than one-quarter of the
circumference of the circle, and the point M
(cf. fig. 26) may fall between 0 and A' and not
between O and A. Also P may be below
the line AOA' and not above it, as in fig. 26.
In order to get over this difficulty we have
recourse to the ideas and conventions of coordinate geometry in making our complete
definitions of the sine and cosine. Let one
arm OA of the angle be the axis OX, and
produce the axis backwards to obtain its
negative part OX'. Draw the other axis