I know they are the same thing, but my book (Schaum's Outline of Trigonometry) has a small section for "Circular Functions" where it says, among other things, that:
Each arc length s determines a single ordered pair (cos s, sin s) on a unit circle. Both s and cos s are real numbers and define a function (s, cos s) which is called the circular function cosine. Likewise, s and sin s are real numbers and define a function (s, sin s which is called the circular function sine. These functions are called circular functions since both cos s and sin s are coordinates on a unit circle. The circular functions sin s and cos s are similar to the trigonometric functions sin and cos in all regards, since any angle in degree measure can be converted to radian measure, and this radian-measure angle is paired with an arc s on the unit circle. The important distinction for circular functions is that since (s, cos s) and (s, sin s) are ordered pairs of real numbers, all properties and procedures for functions of real numbers apply to circular functions.
In any application, there is no need to distinguish between trigonometric functions of angles in radian measure and circular functions of real numbers.
I've always been rather thick when it comes to trigonometry, as well as deciphering mathematical definitions, and I have never heard of this method of specifying trigonometric functions by arcs. Could anyone tell me when it is used and perhaps clarify it for me a little?