The standard "geometric" definition for:
is when is "measured" in radians. So we have good properties, for example:
The only time you should use degrees is when the problem specifically gives angles in degrees. For problems in which sine (or cosine) is used just as a function always use radians.
(There are a number of ways to define sine and cosine that do not involve angles at all. So strictly speaking the argument is not in "radians" or "degrees". However, to be able to identify those functions with the sine and cosine functions we learned in trigonometry, we use radians.)
The point that I think you are missing is that Radians are a pure number! They are defined as the arc length of a circle divided by the radius of the same circle so strictly speaking the "units" (length) reduce out leaving a Real number.
Radian - Wikipedia, the free encyclopedia