It doesn't really change Euler's formula. It just adds a factor for x to it (or an additional term to hide inside sin and cos as functions). That's less convenient, not less true.
replyHowever, I don't really see a benefit gained out of doing this to keep your theta between -1 and 1 instead of between -2pi and 2pi. Like it's not a difficult thing to estimate or convert in your head to get a close enough estimation.
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).
replyIf 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.
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)
replyI 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.
replyhttps://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.
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?
replyI think it's Tau[0].
replyThe naming is irrelevant here. The point is that sin(x) ~ x for small x, whereas sinrev(x) ~ rev * x for small x, which is much uglier. And similar things happen to the derivative of sinrev() vs regular sin() and so on. So switching to preferring to express angles in revs instead actually complicates most formulas, at least in some domans.
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