Why a Trig Ratio Depends Only on the Angle

Take any right triangle with a 35 degree angle in it. Double every side, triple every side, shrink it to a tenth its size — the tangent of that 35 degree angle comes out identical every time. That seems strange until you remember what makes two triangles similar: same angles, proportional sides. If triangle B is triangle A scaled by some factor k, then every side of B is k times the matching side of A. Tangent is opposite over adjacent. In triangle A that is a/b. In triangle B it's (k·a)/(k·b). The k's cancel. Same for sine and cosine. The ratio only ever depends on the angle that produced it, never on how big the triangle happens to be.

This answers the misconception that trips people up: sine looks like it's built from two lengths, so why isn't it a length? Because it's a quotient, not a measurement. Dividing a length by a length leaves a pure number with no units — a fraction describing the triangle's shape, not its size. That's exactly why one trig table works for every right triangle on the planet that shares an angle.

This fact is the reason trig ratios are worth defining at all. It's also what lets a fixed angle carry over to the Law of Sines and Law of Cosines later, where triangles no longer have a right angle: the ratios still depend only on angle, just packaged differently. And it's the backbone of the two special right triangles (45-45-90 and 30-60-90), where the side ratios are fixed numbers precisely because the angles are fixed.