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