alt Hacker NewsThe math is definitely not fine with turns, because your Euler formula e^ix = cos x + i sin x no longer holds. We can use a base other than e, namely B = e^2pi which around 535.4916. This doesn't have the nice e properties like d/dx e^x = e^x.
The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's formula, shows that radians are special: like what binary is to computers.
The natural logarithm being its own derivative is in fact directly linked to the derivative a radians-based sin(x) being cos(x) and so on. Make it any other unit, and you have a mess of conversion factors worse than 2pi.
Imagine complex chained derivatives, double and triple derivative, chain and product rules, all stuffed with trig functions and generating gratuitous piles of cascaded conversion constrants because radians were not used.
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I can only imagine what a ridiculous problem it would be to try to re-do, for example, the Vincenty formula for distance between two latitude/longitude points on an oblate spheroid (the earth) if it couldn't use radians.
https://en.wikipedia.org/wiki/Vincenty%27s_formulae
https://www.johndcook.com/blog/2018/11/24/spheroid-distance/
Further, inverse vincenty is pretty much an essential in anything that needs to find the azimuth between two points on a map. Such as for microwave radio link planning purposes.
Karney (2013) is also radian dependent.
For graphics rendering Euler equation doesnt matter. Colours are 0.0-1.0 and have no relation to reality, but it works. Same with rotations (if we’re not using Quaternikns)
In another comment, I asked why people chose to use the symbol τ over just writing turn or "rev(olution)" (defined to be the constant ≈ 6.28318530718) given how unambiguous the latter is as a name for 2π. And why not just write sinrev() or sinturn(), and leave the symbols sin() and rev (defined to be ≈ 6.28318530718) alone?
Also with radians the differential equation x''''(t)=x(t) has {exp(t), exp(-t), sin(t), cos(t)} as the (real) canonical base for its solution space. And x''(t)=-x(t) gets {sin(t), cos(t)} where they even result from the simplest possible (non-trivial) initial conditions (x(0)=0,x'(0)=1 and x(0)=1,x'(0)=0).
If you look at all the simplest differential equations you can think of, the sin(t)/cos(t) functions in radians are almost inevitable independent from their geometric usage.