Alright, hyperbolic functions the cooler cousins of trigonometric functions that nobody really talks about but are actually pretty ***** important in some circles (like physics and engineering). These functions take an input value x, just like their more famous counterparts, except instead of dealing with angles or radians, we’re working with hyperbolas. Yep, you heard that right the same curves that make up a graphing calculator’s favorite pastime are actually pretty useful in math too!

So let’s start by defining our two main players: sinh (sinus hyperbolicus) and cosh (cosinus hyperbolicus). These functions take an input value x, just like their trigonometric counterparts, but instead of dealing with angles or radians, we’re working with the “hyperbolic” version.

Here’s how you can calculate sinh(x) and cosh(x):
sinh(x) = (e^x e^-x)/2
cosh(x) = (e^x + e^-x)/2

Now, if that looks a little familiar to you, it’s because we’re using the exponential function (e) in both of these definitions. And for those who don’t know, e is just a fancy math constant with an approximate value of 2.71828… but let’s not get into that right now!

So what can you do with hyperbolic functions? Well, they come in handy when dealing with things like exponential growth and decay (which are pretty common in physics and engineering), as well as some other more advanced math concepts. But for our purposes today, let’s just focus on the basics how to calculate sinh(x) and cosh(x).

To do this, we can use a calculator or write out the formula by hand (if you’re feeling fancy). Here are some examples:
sinh(1) = (e^1 e^-1)/2 = approximately 0.84147…
cosh(1) = (e^1 + e^-1)/2 = approximately 1.54306…

And there you have it! Hyperbolic functions in a nutshell. They might not be as well known or popular as their trigonometric counterparts, but they’re definitely worth knowing about if you want to do some serious math (or physics) work. So next time someone asks you what sinh(x) and cosh(x) are all about, now you can impress them with your newfound knowledge!

But wait did we mention that hyperbolic functions have a cooler name than their trigonometric counterparts? That’s right, they’re called “hyperbolics” because they deal with hyperbolas instead of circles. And if you think about it, that makes sense after all, the sine and cosine functions are named for their relationship to circles (sin = opposite/radius and cos = adjacent/radius). So why not give these new guys a name that reflects what they’re all about?

In fact, there’s even an interesting connection between hyperbolic functions and trigonometric functions. If you take the natural logarithm of sinh(x) or cosh(x), you get something called the “inverse hyperbolic sine” (asinh) or “inverse hyperbolic cosine” (acosh). And if that sounds familiar, it’s because those functions are essentially the same as their trigonometric counterparts but with a different input and output.

Hyperbolic functions might not be as well known or popular as their more famous cousins, but they’re definitely worth knowing about if you want to do some serious math (or physics) work. And who knows? Maybe someday hyperbolics will become just as beloved and celebrated as sines and cosines!