While it's quite straightforward to solve problems with multiple planar vectors in equilibrium for 2 variables, I'm having issues with those where I'm asked to minimize or maximize a variable along with two others. For example:
- $\theta$ such that $F_1$ is as small as possible
- $F_1$ and $F_2$ such that our resulting vector $F_R = (3 kN, \angle 0°)$
I've gone ahead and established the equilibrium for the $x$ and $y$ axes but I'm at a loss as to how to continue from there, because I don't know $F_2$ to solve for $\theta$, and because I'm assuming this can be done without calculus, although I might be wrong.
$$\sum F_x=0=-5\cos(30°)+F_2 \cos(20°)+F_1 \cos(\theta) -7\cdot4/5$$ $$\sum F_y=0=5\sin(30°)+F_2 \sin(20°)+F_1 \sin(\theta) -7\cdot3/5$$