15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-38_540_741_169_676}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with parametric equations
$$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$
- Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$
where \(k\) is a constant to be found.
The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
- Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
- Show that \(C\) has Cartesian equation
$$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leq x \leq q$$
where \(q\) is a constant to be found.