Sketch a cycloid and locate a normal intercept

Curve $C$ has $x=\theta-\sin\theta$, $y=1-\cos\theta$, $0\le\theta\le2\pi$.

(i) Show that $dy/dx=\cot(\theta/2)$ and find the gradient at $\theta=\pi$. Describe the tangents as $\theta\to0$ and $\theta\to2\pi$.

(ii) Sketch $C$, clearly showing its features at $\theta=0,\pi,2\pi$.

(iii) For $P$ with parameter $p$, $0<p<\pi/2$, show that the normal at $P$ crosses the x-axis at $(p,0)$.