Sketch the graph of \(y = \arcsin x\) for \(- 1 \leqslant x \leqslant 1\).
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), justifying the sign of your answer by reference to your sketch.
Find the exact value of the integral \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x\).
The infinite series \(C\) and \(S\) are defined as follows.
$$\begin{gathered}
C = \cos \theta + \frac { 1 } { 3 } \cos 3 \theta + \frac { 1 } { 9 } \cos 5 \theta + \ldots
S = \sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \ldots
\end{gathered}$$
By considering \(C + \mathrm { j } S\), show that
$$C = \frac { 3 \cos \theta } { 5 - 3 \cos 2 \theta }$$
and find a similar expression for \(S\).
Section B (18 marks)