Question Statement (from Kleppner and Kolenkow "Introduction to Mechanics"):
Let us change variables as follows:
$\sin u = \sin(θ/2)/ \sin(θ_0/2)$
The motivation for this is that although $ θ$ is periodic, as the pendulum swings through a cycle, θ varies between $−θ_0$ and $ θ_0$. On the other hand, $u$ varies between $−π$ and $+π$.
How come $u$ varies between $−π$ and $+π$ ? If we replace $θ$ with $−θ_0$ and $ θ_0$ in the given equation we get the value ranging from $-1$ to $+1$ i.e. angle ranging from $−π/2$ and $+π/2$.
I was also confused why we could suppose "$\sin u = \sin(θ/2)/ \sin(θ_0/2)$"? Elliptical integral is totally new to me.