(i)[3]
By drawing the graphs of $y = \cosec x$ and $y = x(\pi - x)$ for $0 < x < \pi$, demonstrate that the equation $\cosec x = x(\pi - x)$ has exactly two real solutions in the interval $0 < x < \pi$.
(ii)[2]
Show that $\cosec x = x(\pi - x)$ may be rearranged into the form $x = \frac{1 + x^2 \sin x}{\pi \sin x}$.
(iii(a))[3]
Apply the iterative formula $x_{n+1} = \frac{1 + x_n^2 \sin x_n}{\pi \sin x_n}$ to determine $\alpha$ correct to $2$ decimal places, writing each iterate to $4$ decimal places.
(iii(b))[1]
Deduce $\beta$, giving your answer correct to $2$ decimal places.