To find the value of \(Q\), we resolve the forces and use trigonometric identities. The angles between the forces and the resultant are given as 40° and 80°.

Using the cosine rule for the triangle formed by the forces and the resultant:

\(Q - P \cos 60^{\circ} = 12 \cos 80^{\circ}\)

\(P \sin 60^{\circ} = 12 \sin 80^{\circ}\)

From the first equation, we can express \(Q\) in terms of \(P\):

\(Q \cos 80^{\circ} + P \cos 40^{\circ} = 12\)

\(P \sin 40^{\circ} = Q \sin 80^{\circ}\)

Eliminating \(P\) from these equations gives:

\(Q \cos 80^{\circ} + \frac{12 \sin 80^{\circ} \cos 60^{\circ}}{\sin 60^{\circ}} = 12 \cos 80^{\circ}\)

Solving for \(Q\):

\(Q = 8.91\)