Why are turns not used as the default angle measure?

Question

Why is $2\\pi$ radians not replaced by $1$ turn in formulas? The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?

Answer 1

A turn is an accepted unit to measure an angle.

Radians are just more elegant. Mathematicians love elegance.

See the wikipedia article

https://en.wikipedia.org/wiki/Turn_(angle)

Answer 2

The difference is that both degrees and turns have units. The units of degrees is, of course, degrees. The unit of turns is, of course, turns (or revolutions). But a radian is actually a ratio of the radius and an arc length. So, despite the fact that we call it "radians", it is really something like meters/meters so it really has no unit of measure. Consequently, when we use an equation, for example the s = rθ, and angle measured in radians does not change the unit that the radius is measured in (resulting in an arc length of meters). Perhaps a better example is the Physics equation linear velocity (V) equals radius (r) times the angular velocity (ω) or V = rω. The units of r can be meters. The units or ω can be radians/sec. Multiplying them, the "radians" just disappears because it's not really a unit, and the answer becomes meters/sec (a normal measure of linear velocity). If you had the ω in units of turns/sec then the multiplication of that and meters would result in meter-turns/sec and I have no idea what that is.

Clear as mud?