Trigonometric Coincidence I Know that using Taylor Series, the formula of $\sin x$ is
$$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$
and the unit of $x$ is radian (where $\pi/2$ is right angle).
However, the ratio of the circumference and the diameter of a circle is also $\pi$. Is it a coincidence? Or is there a proof?

I'm not sure why you started with the Taylor series as it does not seem relevant to your question.
To clarify:
$\pi$ is defined as the ratio of the circumference of a circle to its diameter.
$$\pi = \frac{circumference}{diameter}$$ 
and the radian is defined as the angle subtended by an arc on the circumference of a circle with length equal to the radius.
There are of course other measurements of angle such as the degree defined as the angle subtended by $\frac{1}{360}$ of the circumference of a circle.  Its thought that 360 was chosen because it can easily be divided by 2, 3, 4, 5, 6, 8, 9, 10 and 12.
Another measure of angle is the grad though it's use is less common than the other two. 
