8. A curve \(C\) has equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = \arcsin \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2 \quad - \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }$$
- Sketch \(C\).
- Given \(x = 2 \sin y\), show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { A - x ^ { 2 } } }$$
where \(A\) is a constant to be found.
The point \(P\) lies on \(C\) and has \(y\) coordinate \(\frac { \pi } { 4 }\)
- Find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
(3)