(i)[2]
By sketching an appropriate pair of graphs, show that the equation $\cosec \frac{x}{2} = \frac{1}{3}x + 1$ has one root in the interval $0 < x \leq \pi$.
(ii)[2]
Show by calculation that this root lies between $1.4$ and $1.6$.
(iii)[2]
Show that, if a sequence of values in the interval $0 < x \leq \pi$ defined by the iterative formula $x_{n+1} = 2 \sin^{-1}\left( \frac{3}{x_n + 3} \right)$ converges, then its limit is the root of the equation in part (i).
(iv)[3]
Use this iterative formula to find the root correct to 3 decimal places. Write each iterative result to 5 decimal places.