[dahart]

> radian is not an arbitrary unit but a dimensionless one

This is by convention, but has been and is still being debated because calling it dimensionless causes some problems. https://en.wikipedia.org/wiki/Radian#Dimensional_analysis

Furthermore, the whole reason to treat radians as dimensionless, the problem, is with angles, not with radians specifically. Degrees are also considered dimensionless. So, a turn could be treated as dimensionless too, with a conversion constant to radians & degrees, just like between degrees and radians.

Of course, the declared dimensionlessness of angles like radians isn’t something generally discussed in pre-college trig courses, that’s a subtle subject that matters more in physics. In my high school trig, we all understood radians to be a unit of angle and never pondered whether angles had dimension.

Also subtle point, but dimensionless doesn’t mean unitless. It’s another separate convention to drop the units when working with radians.