Equivalence of Definitions of Cosine of Angle

Theorem
Let $\theta$ be an angle.

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

Consider the triangle $\triangle OAP$.

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

From Parallelism implies Equal Alternate Angles:
 * $\angle OPA = \theta$

Thus:

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

Definition from Circle implies Definition from Triangle
Let $\cos \theta$ be defined as the length of $AP$ in the triangle $\triangle OAP$.

Compare $\triangle OAP$ with $\triangle ABC$ in the diagram above.

From Parallelism implies Equal Alternate Angles:
 * $\angle OPA = \theta$

We have that:
 * $\angle CAB = \angle OPA = \theta$
 * $\angle ABC = \angle OAP$ which is a right angle

Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAP$ and $\triangle ABC$ are similar.

By definition of similarity:

That is:
 * $\dfrac {\text{Adjacent} } {\text{Hypotenuse} } = \cos \theta$