8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c21767d7-7331-47f7-8e59-06a0727c67c5-13_771_1036_260_593}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve \(C\) with polar equation
$$r = 1 + \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The tangent to the curve \(C\) at the point \(P\) is perpendicular to the initial line.
- Find the polar coordinates of the point \(P\).
The point \(Q\) lies on the curve \(C\), where \(\theta = \frac { \pi } { 3 }\)
The shaded region \(R\) is bounded by \(O P , O Q\) and the curve \(C\), as shown in Figure 1 - Find the exact area of \(R\), giving your answer in the form
$$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$
where \(p , q\) and \(r\) are integers to be found.