Question

F1 sin 30 + F2 sin 80 = M g

F1 cos 30 - F2 cos 80 = 0

How would I solve for F1 and F2 here?

Answer 1

F1 sin 30 + F2 sin 80 = M g .....(1)

F1 cos 30 - F2 cos 80 = 0 .....(2)

Rearrange equation (2) to give:
F1=F2cos(80)/cos(30)....(2A)
Substitute (2A) into equation (1) to give
(F2cos(80)/cos(30))sin(30)+F2sin(80)=Mg ...(1A)

Solve (1A) for F2, and substitute the value of F2 in (2A) to get F1.

I get F1=0.18Mg, and F2=0.92Mg.
Check my calculations.