So we have a force of unknown magnitude acting on these struts at an angle θ measured from strut AB.
The component of the force acting along AB is 600lb, and the magnitude of the force acting along BC is 500lb.
If Φ = 60°, what is the magnitude of F and the angle θ?
Fcos(θ) = 600lb
The Attempt at a Solution
Ok. So, it'll probably help if I knew the third angle of the triangle formed.
180° = ɣ + (60° + 45°)
180° - 105° = ɣ
75° = ɣ
Great. So, I know that Fcos(θ) = 600lb, and Fcos(75° - θ) = 500lb
hm. Fcos(θ)/600 = 1 = Fcos(75 - θ)/500
500Fcos(θ) = 600Fcos(75° - θ)
5Fcos(θ) = 6Fcos(75° - θ)
5cos(θ) = 6cos(75° - θ)
0 = 5cos(θ) - 6cos(75° - θ)
Originally I tried finding where z = 5cos(θ) - 6cos(75° - θ) intersected with z = θ + η where η = 75° + θ, but I couldn't get Wolfram Alpha to understand what I was talking about. Here, I see I should have just left η as
75° - θ, but even still, I have to *ask* Wolfram Alpha what θ works for 0 = 5cos(θ) - 6cos(75° - θ) when
0<=θ<=75° (it gives me an angle of ~30.7°).
Worse, since I couldn't figure it out, I figures if I gave in on the magnitude of F, I could still find the angle. Mastering Engineering told me F = 870lb. So, if Fcos(θ) = 600, θ=arccos(600/F) and arccos(600/870) ≈ 46.4°.
Which was wrong. The θ it wanted was ~34°
Which means everything I did was wrong.
So what triangle magic do I do to get from the initial problem, to the final F=870 θ=34°, without doing something so complicated I need Wolfram Alpha to crunch it out?