Those degrees are a bit awkward - mind providing a picture? It's a bit difficult to decipher what you are trying to really say. The object hanging at 73 degrees doesn't give much help either.

Edit: Under the information provided, I attempted the problem and was provided with the solution of m = .720 kg, and a weight of 7.05 N.

My steps are shown below.

1. sum of all forces in x direction = ma -cos(68)T1 + cos(38)(2.75) = 0 // -T1x + T2x = 0 (since rest) T1 = 5.78 N

2. sum of all forces in y direction = ma (5.78)sin(68) + (2.75)sin(38) - m(9.8) = 0 // T1y + T2y - weight = 0 (since at rest) m = .720 kg

The angles in the problem don't make a lot of sense. If the left and right string angles are measured from the same point, then their angles put them both in the same quadrant which means the object can't be at equilibrium. And those can't be measurements from vertical because that'd put them below the object, where they couldn't support it. As far as the 73 degrees, it sounds like that's meant to tell us something about the tension on one of the strings, but Nal_rA's interpretation of the information as-is does suggest that's extraneous. Without a diagram I don't think you can get a reliable answer.

Edit: Under the information provided, I attempted the problem and was provided with the solution of m = .720 kg, and a weight of 7.05 N.

My steps are shown below.