If you have a chain suspended between two points at an equal height, the curve it forms should be a catenary:
$$ y = a\cosh\frac{x}{a} $$
The chain I'm interested in is suspended between two points at the same height, with $2x_s$ meters being the distance between them. I also know that the middle of the chain hangs $s$ meters below the suspension points.
What I wish to know is at what angle does the chain connect to the suspension points? By differentiating the catenary, I was able to get this as the equation of the angle:
$$ \alpha = \arctan\left(\sinh \frac{x_s}{a}\right) $$
The issue is, I don't know how to calculate $a$ given the information I have. I got to this equation by subtracting suspension points and the lowest point:
$$ a\cosh \frac{x_s}{a} - a - s = 0 $$
Does this have a sane analytical solution or do I have to calculate it numerically?