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
$$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$
respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
Find the polar coordinates of \(P\).
In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\).
Find, in exact form, the area of \(R\).