7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2026c49f-243b-497a-b702-e40d012ad308-20_465_1070_255_507}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with polar equation
$$r = 4 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 } \text { and } \frac { 3 \pi } { 4 } \leqslant \theta \leqslant \frac { 5 \pi } { 4 }$$
The lines \(P Q , Q R , R S\) and \(S P\) are tangents to \(C\), where \(Q R\) and \(S P\) are parallel to the initial line and \(P Q\) and \(R S\) are perpendicular to the initial line.
- Find the polar coordinates of the points where the tangent SP touches the curve. Give the values of \(\theta\) to 3 significant figures.
- Find the exact area of the finite region bounded by the curve \(C\), shown unshaded in Figure 1.
- Find the area enclosed by the rectangle \(P Q R S\) but outside the curve \(C\), shown shaded in Figure 1.