By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation
$$\operatorname { cosec } x = x ( \pi - x )$$
has exactly two real roots in the interval \(0 < x < \pi\).
Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
(a) Use the iterative formula
$$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$
to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
(b) Deduce the value of \(\beta\) correct to 2 decimal places.