A particle \(P\) 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 \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac{4}{5} \sqrt{5 a g}\) perpendicular to the string and just completes a vertical circle (see diagram).

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