- The curve \(C\), in the standard Cartesian plane, is defined by the equation
$$x = 4 \sin 2 y \quad \frac { - \pi } { 4 } < y < \frac { \pi } { 4 }$$
The curve \(C\) passes through the origin \(O\)
- Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the origin.
- Use the small angle approximation for \(\sin 2 y\) to find an equation linking \(x\) and \(y\) for points close to the origin.
- Explain the relationship between the answers to (a) and (b)(i).
- Show that, for all points \(( x , y )\) lying on \(C\),
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { a \sqrt { b - x ^ { 2 } } }$$
where \(a\) and \(b\) are constants to be found.