9 It is given that \(\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3\), where the constant \(a\) is such that \(0 < a < \frac { 3 } { 2 } \pi\).
- Show that \(a\) satisfies the equation
$$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
- Verify by calculation that \(a\) lies between 2.5 and 3 .
- Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.