Bob?
Now, I like that answer.
The correct answer was given by John Downey.
The name of the curve is a Catenary. Here are two ways to state the formula, one is as John said: Y = a*cosh(x/a). The other is a function of e**x: Y = a/2 * (e**(x/a) + e**(-x/a).
cosh stands for the
hyperbolic cosine of (x/a). In differential equations, the hyperbolic functions were just briefly mentioned. The Electrical Engineers get into them in depth.
Interestingly, the Gateway to the West Arch in St. Louis is an inverted catenary. The formula is given inside the arch.
OK, I didn't remember all this from 45 years ago, but I did remember that a wire between poles form a catenary, and that the formula for it was a function of e**x. Strange what you remember from a diffy-q course.
The other questions were tricky also.
1/sin x = csec x (cosecant)
1/cos x = sec x (secant)
I don't think I have used a secant function since college.
The other day I said that your automobile does calculus before there were computers in cars. For those that were curious and those that don't give a roaring rats rear, here is the story.
The calculus machine in your car is your speedometer and odometer.
From a stop, the speedometer measures acceleration(a/sec*sec) over a period of time, and integrates that to obtain your velocity in miles/hour, or feet/sec or any other convenient measure. The odometer takes the velocity and integrates that into miles driven, or simply miles.
For those of you that think calculus is the stuff you clean off the shower tiles, the integration of a function is to add up all the infinitely small slices along a curve. Adding up the infinitely small snapshots of acceleration at infinitely short periods of time, gives you the velocity at a point of the acceleration versus time curve. Likewise, the odometer does the same operation on velocity to obtain the total distance traveled.
The sun is hurting my eyes, so I will go crawl back under my rock and be quiet.