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 \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(\sqrt{3 a g}\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.

Given that \(\cos \alpha=\frac{4}{5}\), find the value of \(\cos \theta\).