By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } \frac { 1 } { 2 } x = \mathrm { e } ^ { x } - 3\) has exactly one root, denoted by \(\alpha\), in the interval \(0 < x < \pi\).
Verify by calculation that \(\alpha\) lies between 1 and 2 .
Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula
$$x _ { n + 1 } = \ln \left( \operatorname { cosec } \frac { 1 } { 2 } x _ { n } + 3 \right)$$
converges, then it converges to \(\alpha\).
Use this iterative formula with an initial value of 1.4 to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
State the minimum number of calculated iterations needed with this initial value to determine \(\alpha\) correct to 2 decimal places.