>But math never decreed that sine and cosine have to take radian arguments!
Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians.
Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place.
"Turns" don't generalize to higher dimensions either. With radians you can calculate arc length on a circle by multiplying with the radius. This extends naturally to higher dimensions: a solid angle measured in steradians lets you calculate surface area on a sphere by multiplying with the radius. How do you do the same with "turns" on a sphere? You can't in any meaningful way.
Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians.
Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place.
"Turns" don't generalize to higher dimensions either. With radians you can calculate arc length on a circle by multiplying with the radius. This extends naturally to higher dimensions: a solid angle measured in steradians lets you calculate surface area on a sphere by multiplying with the radius. How do you do the same with "turns" on a sphere? You can't in any meaningful way.