Length of Arch of Sine Function

Theorem
The length of one arch of the sine function:
 * $y = \sin x$

is given by:
 * $L = 2 \sqrt 2 E \left({\dfrac {\sqrt 2} 2, \dfrac \pi 2}\right)$

where $E$ denotes the complete elliptic integral of the second kind.

Proof
Let $L$ be the length of one arch of $y = \sin x$.

Then: