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 lengthsdetermines a single ordered pair (coss, sins) on a unit circle. Bothsand cossare real numbers and define a function (s, coss) which is called thecircular function cosine. Likewise,sand sinsare real numbers and define a function (s, sinswhich is called thecircular function sine. These functions are calledcircular functionssince both cossand sinsare coordinates on a unit circle. The circular functions sinsand cossare similar to the trigonometric functions sin$\displaystyle \theta$and cos$\displaystyle \theta$in all regards, since any angle in degree measure can be converted to radian measure, and this radian-measure angle is paired with an arcson the unit circle. The important distinction for circular functions is that since (s, coss) and (s, sins) 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?

Thank you.