7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d3e1c8e-c659-4cfe-82ac-5bfce0f58ba3-24_445_597_248_676}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of curve \(C\) with polar equation
$$r = 3 \sin 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
The point \(P\) on \(C\) has polar coordinates \(( R , \phi )\). The tangent to \(C\) at \(P\) is perpendicular to the initial line.
- Show that \(\tan \phi = \frac { 1 } { \sqrt { 2 } }\)
- Determine the exact value of \(R\).
The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and the line \(O P\), where \(O\) is the pole.
- Use calculus to show that the exact area of \(S\) is
$$p \arctan \frac { 1 } { \sqrt { 2 } } + q \sqrt { 2 }$$
where \(p\) and \(q\) are constants to be determined.
Solutions relying entirely on calculator technology are not acceptable.