Prove that the trigonometric ratios are equal for a given angle regardless of the length of the arms of the angle.

Question

Let $\ XOY$ be an angle.

Let $\ A$ and $\ a$ be two points on OX. Drop perpendiculars to OY from $\ A$ and $\ a$ that intersect OY at $\ B$ and $\ b$ respectively. It is required to prove that $\frac{AB}{OA} = \frac{ab}{Oa}$. How to proceed? Drawing I found this in a book but I didn't understand it

Answer 1

Triangles $ OAB, Oab$ are similar due to the fact that $ab$ is parallel to $AB$.

Corresponding angle equal. At $(b,B)$ we have right triangles.

So the trig ratios can be defined either with the small triangle or the bigger triangle.