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

188 INTRODUCTION TO MATHEMATICS
Then the sign of this generalized angle is y
r
and its cosine is -. With these definitions
r
the functional relations v =sin u and w =cos
u, are at last defined for all positive real values
of u. For negative values of u we simply take
rotation of Q in the opposite (clockwise)
direction; but it is not worth our while to
elaborate further on this point, now that the
general method of procedure has been
explained.
These functions of sine and cosine, as thus
defined, enable us to deal with the problems
concerning the triangle from which Trigonometry took its rise. But we are now in a
position to relate Trigonometry to the wider
idea of Periodicity of which the importance
was explained in the last chapter. It is easy
to see that the functions sin u and cos u are
periodic functions of u. For consider the
position, P (in fig. 27), of a moving point, Q.
which has started from A and revolved round
the circle. This position, P, marks the angles
arc AP arc AP arc AP
--—, and & 2 +,and 4 7r +
r r r
arc AP
and 6 r + ar, and so on indefinitely.
r
Now, all these angles have the same sine and
cosine, namely, Y and x. Hence it is easy to