A turn is an accepted unit to measure an angle.

Radians are just more elegant. Mathematicians love elegance.

See the wikipedia article

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?

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A turn is an accepted unit to measure an angle.

Radians are just more elegant. Mathematicians love elegance.

See the wikipedia article

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.

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