Hyperbolic Functions
Definitions and properties of sinh, cosh, and tanh hyperbolic functions.
Learn their formulas, identities, and applications with examples.
The Definitions
cosh(x) = (ex + e−x) / 2
tanh(x) = sinh(x) / cosh(x) = (ex − e−x) / (ex + e−x)
Hyperbolic functions are analogs of the regular trigonometric functions, but based on hyperbolas instead of circles. They are defined using the exponential function ex rather than angles.
Despite their abstract-sounding name, hyperbolic functions appear naturally in many physical situations: the shape of a hanging cable (catenary), the velocity addition formula in special relativity, and solutions to certain differential equations.
Key Properties
| Function | Domain | Range |
|---|---|---|
| sinh(x) | All real numbers | All real numbers (−∞, +∞) |
| cosh(x) | All real numbers | [1, +∞) |
| tanh(x) | All real numbers | (−1, +1) |
Fundamental Identity
This is the hyperbolic analog of the Pythagorean identity sin²(x) + cos²(x) = 1. Note the minus sign — this reflects the hyperbolic (rather than circular) nature of these functions.
Other Useful Identities
- sinh(−x) = −sinh(x) (odd function)
- cosh(−x) = cosh(x) (even function)
- sinh(2x) = 2 sinh(x) cosh(x)
- cosh(2x) = cosh²(x) + sinh²(x)
- d/dx sinh(x) = cosh(x)
- d/dx cosh(x) = sinh(x)
Example 1
Calculate sinh(2) and cosh(2), then verify that cosh²(2) − sinh²(2) = 1.
sinh(2) = (e² − e⁻²) / 2 = (7.389 − 0.135) / 2 = 7.254 / 2 = 3.627
cosh(2) = (e² + e⁻²) / 2 = (7.389 + 0.135) / 2 = 7.524 / 2 = 3.762
Check: cosh²(2) − sinh²(2) = (3.762)² − (3.627)² = 14.153 − 13.155
= 0.998 ≈ 1 (the small error is from rounding — the identity holds exactly)
Example 2
A hanging cable (catenary) has the equation y = a × cosh(x/a). If a = 10 meters, what is the height of the cable at x = 5 m and x = 15 m above the lowest point?
At x = 5: y = 10 × cosh(5/10) = 10 × cosh(0.5)
cosh(0.5) = (e0.5 + e−0.5) / 2 = (1.6487 + 0.6065) / 2 = 1.1276
y = 10 × 1.1276 = 11.276 m (cable hangs 1.276 m above the lowest point)
At x = 15: y = 10 × cosh(1.5) = 10 × 2.3524 = 23.524 m
At x = 5 m: height = 11.28 m. At x = 15 m: height = 23.52 m (the catenary curve rises steeply away from center).
When to Use These
Hyperbolic functions appear in many areas of math, physics, and engineering.
- Describing the shape of hanging cables and suspension bridges (catenary curves)
- Solving certain types of differential equations
- Special relativity (rapidity and velocity addition)
- Modeling signal attenuation in transmission lines
- Laplace transforms and control systems engineering