Did I solve this correctly? (Calculating forces at rest)

1. The problem statement, all variables and given/known data

A beam (red line) has mass m and length L.
3 chains are attached to the beam to keep it at rest. C1 which is perpendicular to the roof, C2 which is perpendicular at the center of mass on the beam, and C3 which is perpendicular to the side wall.

Questions:
Draw a free body diagram.
Calculate the forces from the 3 chains.

2. Relevant equations
Newton’s law, torque.

3. The attempt at a solution
FBD:

To find the forces from the 3 chains, I set up 3 equations. Using Newton’s 1st law for forces on the x-axis and again Newton’s 1st law for forces on the y-axis:

[itex]\rightarrow \sum F_x=0: C2 \cdot \cos \theta – C3=0[/itex]

[itex]\uparrow \sum F_y=0: C1 – C2 \cdot \sin \theta – mg=0[/itex]

For the 3rd equation, I use the torque on southern most point of the beam (where C3 is attached) Counter-clockwise:

[itex]C1 \cdot L \sin \theta – C2 \cdot \frac{L}{2}=0[/itex]

Now I have 3 equations which is enough to find the values for C1, C2, and C3.
What do you guys think of my method? Did I do this correctly?

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