The tension in the string is the same on either side of the ring, and acts upon the ring in a direction along each section of the string. So the ring is in equilibrium under the action of four forces:
- its weight vertically downwards
- the force horizontally
- a tension, say, inclined at an angle above the horizontal
- a tension inclined at an angle below the horizontal
Resolve vertically, and you can find directly without needing to involve .