A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
Sketch the curve.
Find the area of the region enclosed by the curve, giving your answer in exact form.
Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians).
Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).