A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle moves in complete vertical circles with centre \(O\), with the string taut.

When the string makes an angle \(\theta\) with the downward vertical through \(O\), the speed of \(P\) is \(\sqrt{4ag}\). The ratio of the greatest and least tensions in the string during the motion is \(11:1\).

Find the value of \(\cos\theta\).