Er the question seems pretty straightforward.
The ring in question is in equilibrium. So the resultant of all the forces on it has to be zero.
So the components of F1, F2, F3 in X-Direction Y-Direction should add in such a way as to result in zero.
The component of the forces in X direction is merely the magnitude multiplied by the cosine of the angle made by the forces with POSITIVE X-axis.
component of F1 = F1cos(30) = sqrt(3)F1/2
Component of F2= F2cos(90+45) =F2cos(135)= -F2/sqrt(2)
Component of F3 =F3cos(90+180)=F3cos(270)=0
Now, for equilibrium
Hence sqrt(3)F1/2 = F2/sqrt(2)........(1)
Similarly find the components of F1, F2,F3 along Y- direction.
It is the magnitude multiplied by the sine of the angle made by the force with the X-axis
Components are F1sin(30), F2sin(135) and F3sin(270)
i.e. F1/2, F2/sqrt(2), -F3
Again, these components should add to give zero. Thus you obtain another equation in F1 and F2 as F3 is known.
Solving them simultaneously will yield F1 and F2.