7. (a) Sketch the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)
Show the coordinates of the points where the graph crosses the \(x\)-axis.
The table below gives corresponding values of \(x\) and \(y\) for \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
The values of \(y\) are rounded to 3 decimal places where necessary.
| \(x\) | 0 | \(\frac { \pi } { 8 }\) | \(\frac { \pi } { 4 }\) | \(\frac { 3 \pi } { 8 }\) | \(\frac { \pi } { 2 }\) |
| \(y\) | 0.5 | 0.793 | 0.966 | 0.991 | 0.866 |
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin \left( x + \frac { \pi } { 6 } \right) \mathrm { d } x$$
Give your answer to 2 decimal places.