[bmacho]

The sine and cosine that are defined with Taylor series are not the same sine and cosine that are defined for right triangles.

The former are R->R functions, while the latter are defined on Angles (Angle is unfortunately not an SI physical dimension yet, but I expect it soon to change), and they don't care about the measurement unit.

I have no idea what you mean by radians generalizing for higher dimensions, but not turns.

[gus_massa]

In 2D you can measure the solid angles using steradians.

I guess that turns interpreted as parts of whole circles generalize to parts of whole spheres, and you should divide by 4pi instead of 2pi???

[jameshart]

sine and cosine are functions from ℝ->[-1,1]. They don't take in a value which has a unit, or even a dimension, they take in a real number.

sin(x) is precisely the unique function f(x) such that f''(x) = -f(x). Similar to how exp(x) is the unique function g(x) such that g'(x) = g(x).

Sine does not operate on 'angles measured in radians'. It operates on real numbers. It is zero whenever the real number passed in is a multiple of pi. It happens to have applications in relating angles to distances in circles and triangles, and in order to use sine in that context it is useful to introduce the concept of a 'radian' as a specific, constructed angle of a particular size, such that when you express an angle in terms of multiples of a radian, you can just use the sine function to generate useful values.