8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ed3689f7-b3f0-447b-baa5-e44b8d8342d0-28_522_1084_260_495}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The curve \(C\) shown in Figure 1 has polar equation
$$r = 1 - \sin \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The point \(P\) lies on \(C\), such that the tangent to \(C\) at \(P\) is parallel to the initial line.
- Use calculus to determine the polar coordinates of \(P\)
The finite region \(R\), shown shaded in Figure 1, is bounded by
- the line with equation \(\theta = \frac { \pi } { 2 }\)
- the tangent to \(C\) at \(P\)
- part of the curve \(C\)
- the initial line
- Use algebraic integration to show that the area of \(R\) is
$$\frac { 1 } { 32 } ( a \pi + b \sqrt { 3 } + c )$$
where \(a\), \(b\) and \(c\) are integers to be determined.