That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on.

It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.

Of course that assumes that arclength is well-defined. The standard approach to which is, of course, Calculus.

If you don't have calculus, you don't have anything like a delta-epsilon proof of continuity. But without calculus, you also don't know that you need it. So you just assume (correctly) that you can interpolate, and it works just like you expect, and life goes on.

But even a disembodied being of pure reason might eventually discover continuity via logic->topology.