I ran into a very practical problem math building a gate this weekend. Basically I had two 2x4s running parallel a certain distance apart, and I wanted to cut a cross piece that for aesthetic reasons ran from one end to the other. This is obvious a simple SOHCAHTOA for a line, but for a beam of finite width it isn’t so simple. I ended up solving it practically (physically do it, and measure), but now I can’t make the math work and it is bothering me.

I end up with an equation, but it has two trig functions that I can’t reduce with a trig identity, so it isn’t directly solvable by an inverse trig function. It seems like it should have an obvious direct solution…