Well... you can use the half-angle formulas and the angle addition formulas to calculate sin and cos for angles that are arbitrarily close to the ones that you want. Add to that the idea that sin and cos must be continuous (I consider that intuitively obvious from a unit circle, but I don't know how to make that argument rigorous), and you can start to interpolate. You can in fact use these methods to calculate sin and cos for an arbitrary angle to any desired degree of precision... if you have the patience. It will be shorter to use the series derived from calculus, I'll admit.
I do not believe that there is an argument for continuity without starting with Calculus. Certainly starting from ruler and compass constructions it is not obvious.
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.
It's easy to show more or less directly (by comparing arclength to straight-line length), and certainly without calculus, that the absolute difference between sin(x + delta) and sin(x) is at most |delta|.
Replying really late, just in case anybody reads this. (I went on vacation, and this occurred to me then.)
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.
