Throwing

A projectile is thrown, idealized (no resistance, constant gravity, etc).  What angle sends it the "farthest"?  If "farthest" means that it lands as far away as possible, the answer is a simple Calculus exercise: 45 degrees.  But if "farthest" means the total arc length traveled, the answer oddly enough is the solution x to csc(x) = coth(csc(x)), about 0.9855 radians or 56.47 degrees.  Joint work with Anton Strizhov.