To find \(\theta\) and \(P\), we resolve the forces horizontally and vertically, and use Lami's theorem.

Step 1: Resolve Vertically

Using vertical equilibrium: \(2P \sin \theta = P \sin 60\)

\(\Rightarrow 2 \sin \theta = \sin 60\)

\(\Rightarrow \theta = 25.7^\circ\)

Step 2: Resolve Horizontally

Using horizontal equilibrium: \(2P \cos \theta + P \cos 60 = 10\)

Substitute \(\theta = 25.7^\circ\):

\(2P \cos 25.7 + P \cdot 0.5 = 10\)

\(\Rightarrow P = 4.34 \text{ N}\)

Alternative Method: Lami's Theorem

Using Lami's theorem:

\(\frac{2P}{\sin 120} = \frac{P}{\sin(180 - \theta)} = \frac{10}{\sin(60 + \theta)}\)

Solve for \(\theta\) and \(P\) using the above relationships, confirming \(\theta = 25.7^\circ\) and \(P = 4.34 \text{ N}\).