Hyperbolic Functions

Lesson notes

Overview

sinh, cosh and tanh are built from eˣ and describe hanging cables and relativity. This topic is rated intermediate and is number 51 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

The hyperbolic functions sinh, cosh and tanh are built from exponentials, for example cosh x = (eˣ + e⁻ˣ)/2. They are to the hyperbola what ordinary trig functions are to the circle, since cosh²x − sinh²x = 1. Cosh describes the hanging chain, tanh is a smooth S-shaped function widely used in neural networks, and in special relativity hyperbolic functions describe how velocities combine.

Reading the relationship

The expression "cosh x = (eˣ + e⁻ˣ)/2" is a compact summary of the idea. It uses the symbols c, o, s, h, x, e; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and an exponential, which changes by the same factor for every equal step. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Secant, Cosecant, Cotangent”, the topic before it in Mathematics. Once you are comfortable, continue with “Damped Oscillations”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.