This problem is related to my previous question on the generalized Lami's theorem. I would like to see how you solve this problem and compare with my solution. My motivation for this problem is that I have not seen A SINGLE problem of this type on the internet that considers a 4-force system in static equilibrium. All the problems that I have seen consider 3 forces and those that consider 4 never ask for three unknowns, but offer more information in a way that can be solved by vector components. How do you solve this problem using vector components? I apologize for the ugly problem.
Note: The cable for $T_2$ only hangs from the vertical line, NOT the horizontal.
As mentioned in one of the comments, the problem does not have a unique solution. Let's show that. You know how to calculate the case when you have only two wires. So start with saying that the wire with $T_2$ is missing. Let's call the solutions $T_{1,1}$ and $T_{3,1}$. $$T_{1,1}\cos42.18^\circ=T_{3,1}\sin81.06^\circ\tag1$$$$ T_{1,1}\sin42.18^\circ+T_{3,1}\cos81.06^\circ=6.21\tag2 $$ Similarly, for the case where $T_1=0$, the solutions will be $T_{2,2}$ and $T_{3,2}$: $$T_{2,2}\sin77.18^\circ=T_{3,2}\sin81.06^\circ\tag3$$$$ T_{2,2}\cos77.18^\circ+T_{3,2}\cos81.06^\circ=6.21\tag4 $$ So what's the meaning of multiplying one set of equations, say the first two, by a factor $f$ between $0$ and $1$? You will get the tensions is the wires for a weight $f\cdot 6.21N$. The new solutions will be just scaled versions of the original. Then multiply the second set of equations by $1-f$, and use superposition of the forces principle to add all the scaled equations together: $$f\cdot T_{1,1}\cos42.18^\circ+(1-f)\cdot T_{2,2}\sin77.18^\circ=f\cdot T_{3,1}\sin81.06^\circ+(1-f) T_{3,2}\sin81.06^\circ$$ $$f\cdot (T_{1,1}\sin42.18^\circ+T_{3,1}\cos81.06^\circ)+(1-f)\cdot(T_{2,2}\cos77.18^\circ+T_{3,2}\cos81.06^\circ)=6.21$$ Using $T_1=f\cdot T_{1,1}$, $T_2=(1-f)\cdot T_{2,2}$, and $T_3=f\cdot T_{3,1}+(1-f)\cdot T_{3,2}$, you get the component equations for your system of four forces. But the $T_1,T_2,T_3$ with the above form will be a solution of the system for any $f\in[0,1]$. You will need to add another constraint on $f$ (or any of the tensions) in oder to get a unique solution.