These unique identities are what make hyperbolic functions so powerful. In elementary calculus, they can be used as powerful trigonometric substitution tools while in more abstract mathematics, hyperbolic functions can be used in Euler’s formula and beyond.
Any modern child would definitely have seen or played with the catenary curve sometime or another. Yet, the seemingly simple shape has so many complex elements and origins. The next time you see a barrier rope at the airport, perhaps you would benefit from taking note of the intriguing shape.
For your enjoyment, feel free to take a look at a graph of all four shapes previously mentioned(green catenary curve, purple upside down parabola, red unit circle, and blue unit hyperbola) courtesy to Desmos: