5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{a5902f63-b19f-4e37-94b8-35c3b47ab9de-08_497_919_270_635}
\end{figure}
The curve shown in Fig. 1 has parametric equations
$$x = \cos t , y = \sin 2 t , \quad 0 \leq t < 2 \pi .$$
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
- Find the values of the parameter \(t\) at the points where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
- Hence give the exact values of the coordinates of the points on the curve where the tangents are parallel to the \(x\)-axis.
- Show that a cartesian equation for the part of the curve where \(0 \leq t < \pi\) is
$$y = 2 x \sqrt { } \left( 1 - x ^ { 2 } \right)$$
- Write down a cartesian equation for the part of the curve where \(\pi \leq t < 2 \pi\).
5. continued