I remember being introduced to radians by my teacher: "If we met an alien civilization, they wouldn't be using degrees. Why would they divide a circle into 360 parts? There's no mathematical reason to do that. But everyone would know to use an arc length equal to the radius. The radius is universal." That made sense to me. There's nothing arbitrary about the radius!
Or is there? Why not use a unit that's the same length as the diameter? I'll call it the "diametan". So whereas 1 radian is the angle needed to make an arc whose length is as long as the radius, 1 diametan is the angle needed to make an arc whose length is as long as the diameter. (An angle of 1 diametan = 2 radians.)
So... radians, diametans... one sounded as good as the other to me. Until I graphed them, and thought about their derivatives.
Compared to the graph of
sin(radians) (which peaks at π/2), the graph of
sin(diametans) will be squashed (it peaks at π/4). This will mess up its
derivative (and its anti-derivative, and its derivative's derivative, etc), instead of keeping the nice alternations between ±sin and ±cos.
Perhaps this should have been obvious to me; the same problem is to be seen with the graph of
sin(degrees) (but instead of being squashed, it's stretched).
What makes radians special is the same thing that makes
e special: they balance things just right such that derivatives (or anti-derivatives) find their way back to the starting function.
I got on this line of thought by working through the derivation of the Maclaurin series for sin x, and wondering what would change if the trig functions expected diametans instead of radians.
