The case of the disappearing angle units, or "the dangle of the angle"

[ijabbott]

(08-13-2019 01:56 AM)jlind Wrote:  You are confusing dimensionless with unitless and equating them. They're not the same. A plane angle, which is dimensionless, is a scalar value with a unit to reflect a quantity, be it radians, degrees, grads, quadrants, sextants, turns, or some other unit of measure. Dimensionless and unitless are two very different things. There are an enormous number of dimensionless scalars with units of measure. The SI unit for a plane angle is the rad, the abbreviation for Radian. The unit for a solid angle is the sr, the abbreviation for Steradian.

But is the distinction important mathematically or only for engineering purposes? From what I can gather, mathematicians (or at least pure mathematicians) tend to think of the trig functions as purely numeric functions, without units. For example, "trig substitution" may be used to make certain integrals more tractable.