Hyperbolic Functions Cosh(x), Sinh(x) and Tanh(x)

This post discusses how the function f(x) = e^x is used to create the hyperbolic trig functions Cosh(x), Sinh(x), and Tanh(x). Trig students immediately recognize the remarkable similarity between identities for the functions Cos(x), Sin(x), and Tan(x), and identities for the functions Cosh(x), Sinh(x), and Tanh(x). The hyperbolic trig functions have many important applications in many branches of mathematics and science. A couple of great examples are provided later in this post. I find these functions fun and interesting to play with, and I continue to find new ways of looking at and understanding these functions.

I will start the discussion by defining hyperbolic trig functions Cosh(x), Sinh(x), and Tanh(x) in terms of the functions y = f(x) = e^x / 2 and y = f(-x) = e^-x / 2 which are neither even nor odd. My last post discussed some of the properties and characteristics of even and odd functions. I concluded the post by showing how to create an even function and an odd function from any function which is not necessarily even or odd. If you have not read that post, you may want to read it before continuing. In view of the results obtained in my previous post, it follows that Cos(x) and Cosh(x) are even functions, and Sin(x), Sinh(x), Tan(x), and Tanh(x) are odd functions. The text box below gives the definitions of the three main hyperbolic trig functions. Graphs A, B, C, and D show the graphs of f(x), f(-x), Cos(x), Cosh(x), Sin(x), Sinh(x), and Tanh(x). As a reminder, the functions Cos(x), Sin(x), and Tan(x) are periodic, but the functions Cosh(x), Sinh(x), and Tanh(x) are not.

The text box below gives a comparison of some standard trigonometric identities and hyperbolic trig identities. Go online and check out the other hyperbolic trig identities. You will be amazed. The other day I found out that the derivative of an even function is an odd function, and the derivative of an odd function is an even function. Being able to understand and recognize even functions, odd functions, one-to-one functions, and the inverse of a function gives a student a whole new level of mathematical maturity and sophistication.

The textbox below shows the infinite Taylor series expansion of the functions Cos(x), Cosh(x), Sin(x), and Sinh(x). It’s interesting to see how close and yet very different the infinite series expansions of the functions are. Notice that the Taylor series expansion of Cos(x) and Cosh(x) are sums and differences of even functions! Also notice that the Taylor series expansion of Sin(x) and Sinh(x) are sums and differences of odd functions! The function e^x is the sum of even and odd functions, and therefore it’s neither even nor odd.

I find the infinite series expansion of the inverse functions for the circular trig functions and the hyperbolic trig functions very interesting. The similarities are striking. One can deduce whether or not the inverse of a function is an even or odd function by just doing a simple inspection the infinite series expansion of the function.

In doing research for this post, I discovered an interesting relationship between a catenary curve and a parabolic curve. Imagine a piece of chain, rope, or cable that is hanging from its endpoints to form a U-shaped curve. Galileo (1564 – 1642) thought that this U-shaped curve was parabolic. In 1691, the mathematicians Leibniz, Huygens and Johann Bernoulli showed that the U-shaped curve is described by the hyperbolic cosine function. Freely-hanging electric power cables, silk threads on a spider’s web, or suspension bridge cables have the U-shaped catenary curve. The Gateway Arch in St. Louis, Missouri is said to be an inverted catenary. All catenary curves are the result of sliding, stretching, rotating, or reflecting the graph of y = Cosh(x). Shown below are the equations and graphs of three curves. The results speak for themselves.

My next post will do a geometric compare and contrast of the circular functions Cos(x) and Sin(x) with the hyperbolic trig functions Cosh(x) and Sinh(x). I will do this by defining Cos(x) and Sin(x) in terms of the unit circle, and by defining Cosh(x) and Sinh(x) in terms of the unit hyperbola.

I will end this post my showing you the graph of the first eight terms of a Fourier approximation of a square sine wave. You will notice that the graph is the graph of an odd function and the Fourier approximation is the sum of eight odd functions. Those even and odd functions are everywhere!