On Sun Jul 12, 2020 at 23:20, Gordon Klaus wrote:
> (I have wondered, though, why mathematicians don't factor
> 2*pi into the trig functions. It's a distraction to see that constant
> repeated so often.)
>
Maybe because reducing the [0;2pi] interval to [0;1] is an abstraction? Radians
is a distance along the circle boundary.
Elias

On Mon, Jul 13, 2020 at 12:19 AM Elias Naur <mail@eliasnaur.com> wrote:
>
> On Sun Jul 12, 2020 at 23:20, Gordon Klaus wrote:
> > (I have wondered, though, why mathematicians don't factor
> > 2*pi into the trig functions. It's a distraction to see that constant
> > repeated so often.)
> >
>
> Maybe because reducing the [0;2pi] interval to [0;1] is an abstraction? Radians
> is a distance along the circle boundary.
Or is it the other way around? [0;1] (or [0;360]) is an interval that
corresponds better to typical human experience/use, and [0;2pi] is an
abstraction that fits more neatly with the rest of mathematics (you
mentioned arc length; also, the derivatives of trig functions are
simpler when using radians; not to mention complex exponentials).
Actually, I think abstraction can go either way; it just depends on
what is essential in the current context. If the context is the whole
of mathematics, then radians does seem simpler. If the context is an
API for specifying angles, then rotations or degrees is simpler.