Hyperbolic Functions

Definition

Hyperbolic functions are analogues of trigonometric functions defined using the natural exponential function $e^x$: $\sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}$

• How to read: “The hyperbolic sine of x is equal to e to the x minus e to the negative x, all divided by two, and the hyperbolic cosine of x is equal to e to the x plus e to the negative x, all divided by two.”
• Meaning: Hyperbolic sine is the odd part of $e^x$; hyperbolic cosine is the even part — analogues of $\sin$ and $\cos$ for the unit hyperbola.

Why It Matters

They are the “cousins” of trigonometry used to model things like hanging cables (catenaries) and the paths of light in special relativity. These functions are indispensable for civil engineers designing bridges and physicists exploring the nature of spacetime.

Core Concepts

• Hyperbolic Geometry: While trig functions are related to the unit circle ($x^2 + y^2 = 1$), hyperbolic functions are related to the unit hyperbola ($x^2 - y^2 = 1$).

  • How to read: “The equation x squared plus y squared equals one describes a circle, while x squared minus y squared equals one describes a hyperbola.”
  • Meaning: Circular trig parametrizes a circle; hyperbolic trig parametrizes a hyperbola via $(\cosh t, \sinh t)$.

• Fundamental Identity: $\cosh^2 x - \sinh^2 x = 1$.

  • How to read: “The hyperbolic cosine squared of x minus the hyperbolic sine squared of x is equal to one.”
  • Meaning: The hyperbolic Pythagorean identity — mirrors $\cos^2 + \sin^2 = 1$ with a minus sign.

• Symmetry: $\cosh x$ is an even function (like $\cos x$), and $\sinh x$ is an odd function (like $\sin x$).

Connected Concepts

• Function Definition, Function Notation
• Function Domain and Function Range
• Graphs of Functions: Properties