By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) has exactly two roots in the interval \(0 < x < \pi\).
The sequence of values given by the iterative formula
$$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$
with initial value \(x _ { 1 } = 2\), converges to one of these roots.
Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.