What are hyperbolic functions?
Hyperbolic functions are built from the exponential function eˣ. They behave a lot like sin and cos, but they belong to a hyperbola instead of a circle:
sinh x = (eˣ − e⁻ˣ) / 2 cosh x = (eˣ + e⁻ˣ) / 2 tanh x = sinh x / cosh x
The other three are reciprocals: coth x = 1/tanh x, sech x = 1/cosh x and csch x = 1/sinh x. Points (cosh t, sinh t) trace the right half of the hyperbola x² − y² = 1, just as (cos t, sin t) trace the unit circle.
Key facts
- cosh² x − sinh² x = 1, the hyperbolic version of cos² + sin² = 1.
- cosh 0 = 1 and sinh 0 = 0. cosh x is never less than 1.
- tanh x always lies between −1 and 1, which is why it's used as an "activation function" in neural networks.
- Derivatives: d/dx sinh x = cosh x and d/dx cosh x = sinh x (no minus sign, unlike sin and cos).
- Inverses: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) for x ≥ 1, artanh x = ½ ln((1 + x)/(1 − x)) for −1 < x < 1.
Where they show up
A hanging chain or power line takes the shape of a cosh curve, called a catenary; the Gateway Arch in St. Louis is an upside-down one. Hyperbolic functions also appear in special relativity, in the shape of waves in shallow water, in heat flow and in machine learning.
Values at common points
| x | sinh x | cosh x | tanh x |
|---|---|---|---|
| 0 | 0.000000 | 1.000000 | 0.000000 |
| 0.5 | 0.521095 | 1.127626 | 0.462117 |
| 1 | 1.175201 | 1.543081 | 0.761594 |
| 2 | 3.626860 | 3.762196 | 0.964028 |
| 3 | 10.017875 | 10.067662 | 0.995055 |
You can also type sinh(2), cosh(1) or tanh(0.5) into the scientific calculator.