By sketching a suitable pair of graphs, show that the equation
$$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$
has one root in the interval \(0 < x \leqslant \pi\).
Show by calculation that this root lies between 1.4 and 1.6.
Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula
$$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$
converges, then it converges to the root of the equation in part (i).
Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.