For fun, I decided to create a sort of "intuitive" (for me, anyhow) approach to degrees and such. As I can recall, degrees are based on (the Mesopotamians?)'s base $60$ math. I've read that radians are purely arbitrary. So, I thought that a number system could be devised with smaller units.
I know this is hardly practical, but I've committed to writing something on the subject. I have defined a function map from degrees to what I call "quadrains". These are based on the quarters of a circle: $0\kappa=0^\circ;1\kappa=90^\circ;2\kappa=180^\circ;3\kappa=270^\circ;4\kappa=360^\circ=0^\circ=0\kappa$, and intermediate values defined by the aforementioned formula.
Now, I wish to define the trigonometric functions. I really only need to define the $\sin$ and $\cos$ functions, because $\tan, \cot,$ etc. can be derived from $\sin$ and $\cos$. I know that $\sin(\theta) = \frac {\text{Opposite}}{\text{Hypotenuese}}$ and that $\cos(\theta) = \frac {\text{Adjacent}}{\text{Hypotenuese}}$, but how might I define the trig functions for my system? I'd follow the example of how radians define it for theirs, but: I have no idea how they did it, either.
I am aware that $\sin$ etc. can be generalized by a Taylor Series, and, typically, I'd be O.K. with that. Now, however, I wish to give our friend Taylor a break and look for alternatives.
TL;DR Quarter-based "degree" system. Can I haz trig?
Radians are most surely not arbitrary! One radian is the angle subtended by a circular arc whose length is one radius of the circle.
I don't believe that the ratio definitions of trigonometric functions care what the reference angle measures $($that is, since $45^\circ \equiv \pi/4\ $radians, we have $\sin(45^\circ) = \sin(\pi/4))$; just like a falling brick doesn't care if the acceleration due to gravity is given in ft/sec$^2$, or m/sec$^2$.
So yes, you can haz as much trig as you want :)