# The Saint Louis Arch and y=coshx

NPR’s Science Friday had an episode highlighting the mathematics behind the Saint Louis Arch. You can watch a little video on the subject at their website, here.

The shape of the arch is the same shape of a hanging chain, called a catenary. “Catenary” is another name for the hyperbolic cosine function,

$\cosh {x} = \frac {e^{x}+e^{-x}} {2}$

It’s not the world’s most riveting video, but it does highlight this important function that doesn’t get much press in our high school math curriculum. If you watch the video, you’ll learn something about this function, and you’ll learn that the catenary is not only the shape of a hanging chain but also the shape of the most stable arch. For more on the mathematics of the Saint Louis Arch, visit the wikipedia article.

In particular, the Saint Louis Arch has the equation

$y = 693.8597 - 68.7672\cosh {(0.0100333x)}$

Now, of course you want to know about the hyperbolic sine function, too. I’ll let you look it up yourself (or maybe you can take a guess, first?). Then ask yourself some questions you might be dying to know: Which functions are odd/even? What trig identities are associated with these functions? For Calculus students: What is the derivative of each of these functions? What is the power series? And there are some exciting connections with these functions and complex numbers, too. Go play, and tell me what you learn!