If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$? In Terry Tao's textbook Analysis, he defines $\sin x$ as below:

  
*
  
*Define rational numbers
  
*Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences
  
*Define reals as the space of Cauchy sequences of rationals modulo equivalence
  
*Define limits (and other basic operations) in the reals
  
*Cover a lot of foundational material including: complex numbers, power series, differentiation, and the complex exponential
  
*Eventually (Chapter 15!) define the trigonometric functions via the complex exponential. Then show the equivalence to other definitions.
  

My question is how can we obtain the geometry interpretation of $\sin x$, that is, the ratio of opposite side and hypotenuse.

From the series, it is easy to see Euler's forumla,
$$ e^{ix} = \cos(x) + i\sin(x)$$
With more series manipulation, we can obtain the Pythagorean theorem,
$$|e^{ix}| = e^{ix}e^{-ix} = (\cos(x) + i\sin(x))(\cos(x) - i\sin(x)) = \cos^{2}(x) + \sin^{2}(x) = 1$$
Knowing that $\sin(x)$ and $\cos(x)$ have range $[-1,1]$, and are odd and even functions respectively, we see that $e^{ix}$ traces out the unit circle in $\mathbb{C}$. From this, we can extract the geometric interpretation of sine and cosine. 
