6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).
- Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
- Verify by calculation that \(a\) is greater than 1 .
- Use the iterative formula
$$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$
to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.