6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
- Find the exact value of the area of the region bounded by \(C\) and the initial line.
- Show that, at the point on \(C\) furthest from the initial line,
$$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$
and verify that this equation has a root between 0.56 and 0.57 .