[EmilioMartinez]

But the proposal here would not change the circumference to 1, it's about representing angles in another currency. More often that not when you are doing trigonometry you care more about angles than arclength. The issue stems from thinking angles in terms of arc length.

[Aardappel]

The angle needing to be used as a length (to be able to use the same scale as a unit radius) is something that naturally happens in a lot of trig algorithms.

It depends whether you care more about the "UI" (the user of the algorithm gets a more pleasant input range to use) vs how convoluted the implementation is (somewhere deep down an extra multiply by 0.15915 was needed).

All I am saying that "turns" are not universally better, they have downsides too.