Where does the sine of an angle actually come from? Everyone in my class just presses the button

There is a formula, several in fact, and the reason nobody shows you is that they are infinite sums — and infinite sums are usually taught a few years after trigonometry.

The main one says that the sine of an angle x, with x measured in radians rather than degrees, equals:

x  -  x³/6  +  x⁵/120  -  x⁷/5040  +  ...

forever, with the pattern being odd powers, alternating signs, and the denominators being factorials. You can try it right now with a calculator: take x = 0.5, compute those four terms, and you will get 0.4794, which is correct to four decimal places already.

That is genuinely how it is done. It is not an approximation to some other true method — this sum is the sine, in the same way that the triangle definition is.

The catch is that this only converges usefully for small x, which is why the second half of the answer matters.

The practical half, since I do this for a living: a real calculator does not just add up that series, and how it avoids doing so is genuinely clever.

The steps are roughly:

1. Reduce the angle. Sine repeats every full turn and has symmetry within that, so any angle can be folded down into a small range near zero using identities you already know — things like sine of (180 − x) equalling sine of x. This is the step that makes everything else cheap.
2. Evaluate the small angle with a short polynomial. Not the plain series, but a polynomial chosen specifically to minimise the worst error over that small range. Those coefficients are computed once, decades ago, and baked into the chip.
3. Restore the sign and quadrant from step 1.

So the answer to "how does it know" is: it does not know your angle at all. It knows how to turn any angle into a small one, and it knows a handful of numbers that make a short polynomial accurate on small ones.

Older calculators used a different method again, based on repeatedly rotating by fixed pre-computed angles and only ever adding and shifting, because they had no multiplier.

You have asked a genuinely good question and it is worth knowing that you can also build the table yourself with nothing but geometry, which is how it was done for centuries before any of the above existed.

Start from the angles you can construct exactly — 30, 45, 60 — using triangles you can draw. Then use the half-angle identity to get 15, then 7.5, and keep halving. Use the addition identity to combine known angles into their sums. Between those two moves you can reach a dense set of angles, and interpolate between them for the rest.

That is essentially how the first trigonometric tables were built, by hand, over years. Doing five or six steps of it yourself is an afternoon well spent and it makes the calculator feel less like magic.

One thing worth knowing early, because it explains a detail that confuses people later: the series above only works with the angle in radians, not degrees.

That is not a convention chosen to annoy students. Radians are the measure in which the series comes out that clean — in degrees you would need an ugly conversion factor scattered through every term. It is the first example you will meet of a unit being chosen because it makes the mathematics simple rather than because it is convenient for humans.