Forces in equilibrium. Tension, ring, pulley and lots of string [With Picture]

[WilliamA]

Homework Statement

To make it easier on the people who will try to help me, I've scanned the problem:

We are interested in finding the following variables:
Angles a and c
The tension between A and B (T_AB)

Homework Equations

The pulley is a "perfect pulley" no friction
The system is at equilibrium
We consider on side A that the string is attached to the wall, or a hook, it's fixed.
All three cords are attached to a ring situated at position B
There is nothing between A and C, but we know there is 3 meters of distance between the two

The Attempt at a Solution

Alright so I decided to make a Free Body Diagram taking the ring as the subject. It goes like that:

Considering the pulley, I guess we can assume that T_BC is going to to be the same as F_A.

After that I decided to decompose each of the three forces, and since everything is at equilibrium, I've come with two equations (again in picture, going to be easier):

At that point I am stuck because I don't see any more equations involving those three unknown variables(a,c, T_AB) I can do. So I have 3 unknown variables but only two equations.

Could someone please help me out?

 

[gneill]

Have you considered using some of the trig laws? The law of cosines and the law of sines come to mind.