8 The diagram shows a sketch of a curve.
\includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726} The polar equation of the curve is $$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) is the point of the curve at which \(\theta = \frac { \pi } { 3 }\). The perpendicular from \(P\) to the initial line meets the initial line at the point \(N\).

1. 1. Find the exact value of \(r\) when \(\theta = \frac { \pi } { 3 }\).
   2. Show that the polar equation of the line \(P N\) is \(r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta\).
   3. Find the area of triangle \(O N P\) in the form \(\frac { k \sqrt { 3 } } { 128 }\), where \(k\) is an integer.
2. 1. Using the substitution \(u = \sin \theta\), or otherwise, find \(\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta\), where \(n \geqslant 2\).
   2. Find the area of the shaded region bounded by the line \(O P\) and the arc \(O P\) of the curve. Give your answer in the form \(a \pi + b \sqrt { 3 } + c\), where \(a , b\) and \(c\) are constants.
      (8 marks)