Why is $A(\cos x)=x/2$ in the unit circle definition of cosine? In the Spivak's calculus, there is the following definition:

If $0 \leq x \leq \pi$, then $\cos x$ is the unique number in $[-1,1]$
  such that 
$$A(\cos x)=\frac x 2 \text{ and } \sin x=\sqrt{1-\cos^2 x}$$
  $$\text{where } A(x)=\frac{x \sqrt{1-x^2}}{2}+\int_x^1 \sqrt{1-t^2}dt$$

This seems hardly a definition, but a theorem. Where does $x/2$ come from? It is stated that it is the area but why $x/2$ and not just $x$?

Areas contain a one-half factor for the same reason as in a triangle: a circular wedge is narrower at the tip, so area = arc*r/2 (as for the full circle, area is $\frac12 (2\pi r^2)$ but circumference is $2 \pi r$ without the one-half ($2\pi$ is a unit here rather than $\pi$ because it's the full circle).
So, essentially, you are saying that the area is half the arc (in dimensionless units - unit circle), and the area is then expressed with an integral.
Another way of looking at it would be that in the first line, you computed the angle in the polar coordinates,
$$A=\int_0^{x}\int_0^1 r\,dr\,d\phi=\frac {x}{2}$$
and in the second line, you computed exactly the same thing in cartesian coordinates, which has a side-effect of defining $\cos$ and $\sin$ through the transformation between polar and cartesian coordinates.
