Why is $\sin \theta$ just $\theta$ for a small $\theta$? When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?

=== Geometric ===
 
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The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the same length, meaning cos θ is close to 1 and $\theta^2/2$ helps trim the red away.
:
$\cos{\theta} \approx 1 - \frac{\theta^2}{2}$
The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = A*θ, from trigonometry, sin θ = O/H and tan θ = O/A, and from the picture, $ O \approx s $ and $ H \approx A $ leads to:
:$\sin \theta = {O\over H}\approx{O\over A} = \tan \theta = {O\over A} \approx {s\over A} = {{A*\theta}\over A} = \theta$.
Simplifying leaves,
:$\sin \theta \approx \tan \theta \approx \theta$.
