The figure below shows a cross-section $O B C E$ of a car headlight whose reflective surface is modelled in suitable units by the curve with parametric equations
$x=a(\theta-\sin \theta), \quad y=a(1-\cos \theta)$
for $0 \leq \theta \leq 2 \pi$, where $a$ is a positive constant.

(i) Find in terms of $a$
(a) the length of $O E$,


(b) the maximum height of the curve $O B C E$.


(ii) Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\cot \frac{\theta}{2}$.


Point $B$ lies on the curve and has parameter $\beta . T S$ is tangential to the curve at $B$ and $B C$ is parallel to the $x$-axis. Given that $\angle T B C=\frac{\pi}{6}$,

(iii) show that $\beta=\frac{2 \pi}{3}$.