6 The equation \(\cot \frac { 1 } { 2 } x = 3 x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).
- Show by calculation that \(\alpha\) lies between 0.5 and 1 .
- Show that, if a sequence of positive values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( x _ { n } + 4 \tan ^ { - 1 } \left( \frac { 1 } { 3 x _ { n } } \right) \right)$$
converges, then it converges to \(\alpha\).
- Use this iterative formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.