Along with the 'convenient' arclength formula already mentioned s=r*theta, do not forget the equally 'convenient' area formula A=r^2*theta/2.
I would agree with the pedagogy mentioned elsewhere: by the time you are dealing with circular functions, the notion of angle should be replaced by position on the unit circle.
Another view of circular functions and radians (and through this the unitless nature of radians) is a presentation of trig functions I learned in Spivak's Calculus book (decades ago). The sine function is the (theorem) unique solution to the differential equation y''=-y, y(0)=0, y'(0)=1 which has a smallest positive zero (another theorem), defined arbitrarily by the symbol pi.
Adam Stinchcombe
Adirondack Comm Coll
Queensbury NY
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