For example, whenever I search for a proof of the Pythagorean theorem, I get a drawing of a geometric proof, yet we use the Pythagorean theorem to algebraically compute distance between points in an plane. Likewise sine cosine and tangent have geometric definitions, yet we determine their values not by drawing right triangles and measuring but by plugging angles into a function.
I've read about how Cartesian Geometry combined Geometry and Algebra, but how can we be sure that the two are compatible?
In other words, how do we know that we can just take a theorem like the Pythagorean Theorem, proved Geometrically, in the realm of rulers of compasses, and apply it to Algebra, in the realm of numbers?
Geometry seems too empirical. Why is it that just because we draw something on paper and it roughly works out that we assume it's true?