8. The curve \(C\) shown in the diagram above has polar equation
$$r = 4 ( 1 - \cos \theta ) , 0 \leq \theta \leq \frac { \pi } { 2 }$$
At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
- Show that \(P\) has polar coordinates \(\left( 2 , \frac { \pi } { 3 } \right)\).
The curve \(C\) meets the line \(\theta = \frac { \pi } { 2 }\) at the point \(A\). The tangent to \(C\) at the initial line at the point \(N\). The finite region \(R\), shown shaded in
\includegraphics[max width=\textwidth, alt={}]{863ef52d-ae75-450c-9eab-8102804868f5-2_737_561_1395_1329} the diagram above, is bounded by the initial line, the line \(\theta = \frac { \pi } { 2 }\), the arc \(A P\) of \(C\) and the line \(P N\). - Calculate the exact area of \(R\).