The first step in any mechanics problem is to draw a free body diagram. Here is a free body diagram of the problem showing all of the forces on the 600 lbs weight, which we idealize as a point. It shows the force, F, the angles α and β, the tension, T, on the rope, and the weight, w = 600lbs. We then break F and T into vertical and horizontal components. Since the problem says the weight "is supported by the rope and pulley" we assume everything is in equilibrium, which tells us that the sum of all forces is zero. We can break this down to say that the sum of the vertical components of the forces is zero, and the sum of the horizontal components is zero. Therfore, from the vertical components we get:
T*cos(β) + F*sin(α) = 600 lbs
From the horizontal components we get:
T*sin(α) = F*cos(β)
BOTH of these equations must be true at equilibrium. I think this should be enough to get you started on your program. HTH.
By the way, someone (ja72) mentioned above that there are two equations and three unknowns. Unless I'm missing something, I agree. You cannot calculate F and α for various values of β without some other assumption, either about the relationship between α and β (for example we could assume they are symmetrical so β = (π/2) - α ), or about the value of T. Note that if we do assume the are symmetrical, then T = F.
What I would suggest is one of two things:
(a) write the program so that the user specifies α and then the program calculates F for that α and the seven β values that the problem asks for, or
(b) for each of the seven values of β calculate the value of F for several values of α, for example for α = 0, α = 90 (i.e. π/2), α = 45 (i.e. π/4) and α = (π/2) - β.