How does the length of the tangent in this picture relate to $\sin(x)/\cos(x)$?

Question

I understand the justificaton behind the length of $\sin(\theta)$ and $\cos(\theta)$. But I don't understand why AE is the length of $\tan(\theta)$, I understand that it satisfies the pythagorean identities but how exactly is that length related to $\frac{\sin(x)}{\cos(x)}$? I also don't understand the lengths of the other trigonometric functions $\sec(\theta)$, $\csc(\theta)$ and $\cot(\theta)$.

Answer 1

Right-angled triangles with one angle equal to $\theta$ include $\triangle OAE,\,FAO$, each with $\theta$ (the right-angle) at the vertex listed first (second). Now use similarity to $\triangle OCA$.

Answer 2

1. $\angle OAE = \frac{\pi}{2} \ (= 90^\circ)$,
2. $\angle CAE = \angle OAE - \angle OAC = \frac{\pi}{2} - \left(\frac{\pi}{2} - \angle COA\right) = \theta$,
3. So now $\Delta CAE$ and $\Delta COA$ share two of the same angles and so must be similar,
4. Hence the ratio of their sides must be equal, and the result follows.

Answer 3

Consider the intersection point $T$ of line $OA$ with the tangent to the circle at $D$. By definition the length $DT$ is equal to $\tan \theta$ since we have a unit circle.

Now the right triangles $OTD$ and $OAE$ are isometric, by the second congruence case, and therefore, the corresponding edges $DT$ and $AE$ have the same length.