They are equally obscured by tau, or whatever other arc-length you might choose! The meaning does not depend on the definition of a circle but on the properties of the exponential function:
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
As for sin(x), it is a function based on circles and arc lengths, but specific values of arc length do not enter the definition.
Actually, you can go backwards and say that its circular properties define the exponential function. Euler's formula describes the rotation of the unit vector through the imaginary plane.
> As for sin(x)...specific values of arc length do not enter the definition
It's not about the definition, it's about the meaning. Sin(x) is the height of the circle at x radians. And it's super awesome with tau: one tau is full circle, and one period.
Well, if you've already made the jump to understanding negative heights as going below the real line. This is a topic for an introductory course in geometry? When I learned about sine and cosine, it was first defined in terms of SOHCAHTOA!
It's exactly like refactoring code: sure, it does the same thing, but now it's more compact, more concise, more clear, more elegant; the parts of the system all fit together better, and people coming onto the project will be able to learn it faster. If you don't care about those things, then you won't refactor your code, or see the point of τ.