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.

Euler's formula (the complex exponential) and the identification of the complex plane $\mathbb{C}$ with $\mathbb{R}^2$ does that for us. Then the equations $x=\cos\theta,~y=\sin\theta$ define the ordinate and abscissa of points on the unit circle; drawing in the relevant triangles gets you the rest of the way.
From the series definition of $\sin$ and $\cos$, the easiest link with geometry is via the complex exponential series to derive Euler's formula, $\cos\theta+i\sin\theta=e^{i\theta}$. Once we have that, the rest follows (from the background that Tao provides).
