How do similar right triangles lead to the definitions of the trigonometric ratios?

The ratio of two sides of one right triangle is the same of the corresponding sides of any its similar triangles.

Call a, b, c the sides of a right triangle and A, B, C the sides of any of its similar triangles.

Then a/b  = A/B, which means that this ratio is a constant.

The same for a/c = A/C, and b/c = B/C.

Even, the same is true for the inverses: b/a = B/A; c/b = C/B, and d/a = C/A.

Then, you can define a function for every one of these ratios. Those functions are the trigonometric functions.