How do the power-series definitions of sin and cos relate to their geometrical interpretations?

I hope that most of you have either asked yourselves this question explicitly, or at least felt a vague sense of unease about how the definitions I gave in lectures, namely

$latex displaystyle cos x = 1 – frac{x^2}{2!}+frac{x^4}{4!}-dots$

and

$latex displaystyle sin x = x – frac{x^3}{3!}+frac{x^5}{5!}-dots,$

relate to things like the opposite, adjacent and hypotenuse. Using the power-series definitions, we proved several facts about trigonometric functions, such as the addition formulae, their derivatives, and the fact that they are periodic. But we didn’t quite get to the stage of proving that if $latex x^2+y^2=1$ and $latex theta$ is the angle that the line from $latex (0,0)$ to $latex (x,y)$ makes with the line from $latex (0,0)$ to $latex (1,0)$, then $latex x=costheta$ and $latex y=sintheta$. So how does one establish that? How does one even define the angle? In this post, I will give one possible answer to…