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.

In this hint I suggest showing from the power series that if
$$
\sin(x)=\sum_{k=0}(-1)^k\frac{x^{2k+1}}{(2k+1)!}\tag{1}
$$
and
$$
\cos(x)=\frac{\mathrm{d}}{\mathrm{d}x}\sin(x)=\sum_{k=0}(-1)^k\frac{x^{2k}}{(2k)!}\tag{2}
$$
that $\frac{\mathrm{d}}{\mathrm{d}x}\cos(x)=-\sin(x)$ and from there that 
$$
\sin^2(x)+\cos^2(x)=1\tag{3}
$$
Therefore, $(\cos(x),\sin(x))$ lies on the unit circle.
To see that $(\cos(x),\sin(x))$ moves around the unit circle at unit speed, note that $(3)$ implies
$$
\left|\frac{\mathrm{d}}{\mathrm{d}x}(\cos(x),\sin(x))\right|=\left|(-\sin(x),\cos(x))\right|=1\tag{4}
$$
Thus, $(3)$ and $(4)$ say that $(\cos(x),\sin(x))$ moves around the unit circle at unit speed. Note also that $(-\sin(x),\cos(x))$ is at a right angle counter-clockwise from $(\cos(x),\sin(x))$. Therefore, $(\cos(x),\sin(x))$ moves counter-clockwise around the unit circle at unit speed, starting at $(1,0)$. This should be sufficient to show that $\sin(x)$ and $\cos(x)$ are the standard trigonometric functions.
