(a) Show that the curve with Cartesian equation \(\left( x - \frac{1}{2} \right)^2 + y^2 = \frac{1}{4}\) has polar equation \(r = \cos \theta\).

The curves \(C_1\) and \(C_2\) have polar equations \(r = \cos \theta\) and \(r = \sin 2\theta\) respectively, where \(0 \leq \theta \leq \frac{1}{2} \pi\). The curves \(C_1\) and \(C_2\) intersect at the pole and at another point \(P\).

(b) Find the polar coordinates of \(P\).

(c) In a single diagram sketch \(C_1\) and \(C_2\), clearly identifying each curve, and mark the point \(P\).

(d) The region \(R\) is enclosed by \(C_1\) and \(C_2\) and includes the line \(OP\). Find, in exact form, the area of \(R\).