7 The diagram below shows the curve with polar equation \(r = \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{58e9b480-f561-4a28-b911-7d9d6a80e976-3_385_807_1834_260}
- Find the values of \(\theta\) at the pole.
- Find the polar coordinates of the point on the curve where \(r\) takes its maximum value.
- In this question you must show detailed reasoning.
Find the exact area enclosed by the curve.
- Given that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\), find a cartesian equation for the curve.