It is given that, for non-negative integers \(n\),
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$
Show that, for \(n \geqslant 2\),
$$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
The equation of a curve, in polar coordinates, is
$$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$
(a) Find the equations of the tangents at the pole and sketch the curve.
(b) Find the exact area of the region enclosed by the curve.
RECOGNISING ACHIEVEMENT