8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b8b399d-ba16-4fcb-be45-0ba40a7ae09d-13_542_748_205_607}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).
- Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
- Find the exact value of \(R\).
The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
- Use calculus to show that the exact area of \(S\) is
$$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$