Particles \(A\) and \(B\), of masses \(3 m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac{2}{5} \sqrt{g a}\). The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac{12}{5} m g\).
(a) Find \(\cos \theta\).

(b) Find, in terms of \(a\), the distance of \(B\) below the ring.