To elaborate, “angle measure” (i.e. circular arclength corresponding to a particular rotation) is a derived quantity, not the primary abstraction we should be thinking about.

The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm.

Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1.

This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.

>I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products.

What about geometric algebra instead?

http://www.shapeoperator.com/2016/12/12/sunset-geometry/