Angle is technically dimensionless. Not sure how to explain the "zen" of radians; it's kind of the angular equivalent of "e" in natural logarithms. People who

Angle is technically dimensionless. Not sure how to explain the "zen" of radians; it's kind of the angular equivalent of "e" in natural logarithms. People who first learn logarithms say WTF, what is this "e" thing? why not just log10 everywhere? But when you get into calculus you find that "e" is the only log base that doesn't introduce extra baggage; if you take the derivative of a^x the result is ln a * a^x, and if you take the derivative of log_a x = ln x / ln a, you get 1/(x ln a). In both cases the "ln a" term is 1 only when a = e. For radians, the functions sin wt and cos wt (w = "omega") have derivatives of w * cos wt and -w * sin wt when wt is measured in radians. Otherwise there is an additional constant of proportionality that depends on the angular "unit". (the term "unit" in quotes because it's still dimensionless but just related to radians by a dimensionless scaling factor, like 180/pi for radians <-> degrees or 1/(2*pi) for radians <-> revolutions.