Why Similarity Makes the Trig Ratios Work

Take any right triangle with a 40 degree angle. Scale it up, shrink it, it doesn't matter: two angles match (the right angle and the 40 degree angle), and matching two angles forces the third to match too, since angles in a triangle add to 180 degrees. That's angle-angle similarity, and it means all these triangles are the same shape, just different sizes. Similar shapes have proportional sides, so the ratio of any two sides, opposite over hypotenuse, adjacent over hypotenuse, opposite over adjacent, stays fixed no matter which copy you measure. That fixed ratio is what sine, cosine, and tangent actually are.

This is why a 3-4-5 triangle and a 9-12-15 triangle give the same sine for their smallest angle: 9-12-15 is just the 3-4-5 triangle scaled by 3, so the sides grow together and their ratio, 3 over 5 or 9 over 15, comes out to 0.6 either way. The angle is what's fixed; the triangle's size is free to change. That's the key fix for the misconception that sine belongs to a triangle: it belongs to the angle, and any triangle with that angle will report the same value.

The same logic explains slope. A line through the origin creates a family of right triangles, one for every point you pick on it, each with the same angle to the x-axis. Angle-angle similarity makes them all similar, so rise over run is identical for every pair of points, which is exactly the constant slope you compute from any two points on a line.