Hyperbolic Functions
sinh, cosh, tanh
About This Calculator
Hyperbolic functions are analogs of trigonometric functions but defined using the hyperbola rather than the circle. sinh and cosh arise naturally in the equations for hanging cables (catenaries), heat transfer, relativistic velocity addition, and the solution to many differential equations.
Formula
sinh(x) = (eˣ − e⁻ˣ) / 2
cosh(x) = (eˣ + e⁻ˣ) / 2
tanh(x) = sinh(x) / cosh(x) = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
Identity: cosh²(x) − sinh²(x) = 1
Example Calculation
Calculate sinh(1), cosh(1), tanh(1).
- e¹ ≈ 2.7183; e⁻¹ ≈ 0.3679
- sinh(1) = (2.7183 − 0.3679)/2 = 1.1752
- cosh(1) = (2.7183 + 0.3679)/2 = 1.5431
- tanh(1) = 1.1752/1.5431 = 0.7616
sinh(1)≈1.1752, cosh(1)≈1.5431, tanh(1)≈0.7616
Hyperbolic Function Values
| x | sinh(x) | cosh(x) | tanh(x) |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 0.5 | 0.5211 | 1.1276 | 0.4621 |
| 1 | 1.1752 | 1.5431 | 0.7616 |
| 2 | 3.6269 | 3.7622 | 0.9640 |
| 3 | 10.0179 | 10.0677 | 0.9951 |
Frequently Asked Questions
How are hyperbolic and trig functions related?
Trig functions are defined on a unit circle (x²+y²=1); hyperbolic functions on a unit hyperbola (x²−y²=1). They share identical-looking identities but with sign differences, e.g. cosh²−sinh²=1 vs sin²+cos²=1.
What is a catenary?
A catenary is the curve formed by a freely hanging chain or cable under gravity. Its equation is y = a·cosh(x/a), making cosh the natural function for describing sagging cables, suspension bridge main cables, and hanging power lines.
What are inverse hyperbolic functions?
arcsinh(x) = ln(x + √(x²+1)); arccosh(x) = ln(x + √(x²−1)). These appear in integration results and are available in most math libraries as asinh, acosh, atanh.
Where does tanh appear in machine learning?
tanh is a common activation function in neural networks. It squashes input to the range (−1, 1), is zero-centered (unlike sigmoid), and its derivative tanh'(x) = 1−tanh²(x) is easy to compute during backpropagation.