Equivalence of Definitions of Sine of Angle

Theorem
Let $\theta$ be an angle.

Then the following definitions of sine of $\theta$ are equivalent:

Definition from Triangle implies Definition from Circle
Let $\sin \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$ in a right triangle.

Consider the triangle $\triangle OAP$.

By construction, $\angle OAP$ is a right angle.

Thus:

That is:
 * $\sin \theta = AP$

Definition from Triangle implies Definition from Circle
Consider the