Cosh Calculator (Hyperbolic Cosine)

Understanding the Hyperbolic Cosine Function

The hyperbolic cosine function, written as cosh(x), is one of the fundamental hyperbolic functions. Unlike the regular cosine function which relates to the unit circle, the hyperbolic cosine relates to the unit hyperbola. It is defined using the exponential function as cosh(x) = (e^x + e^-x) / 2.

The graph of cosh(x) forms a catenary curve, which is the shape of a hanging chain or cable under its own weight. This appears in architecture (the Gateway Arch), engineering (suspension bridges), and physics (minimal surfaces).

Hyperbolic Function Formulas

Hyperbolic Cosine

The even part of the exponential function. Always greater than or equal to 1.

cosh(x) = (e^x + e^(-x)) / 2

Hyperbolic Sine

The odd part of the exponential function. Ranges from negative infinity to positive infinity.

sinh(x) = (e^x - e^(-x)) / 2

Hyperbolic Tangent

The ratio of sinh to cosh. Ranges between -1 and 1.

tanh(x) = sinh(x) / cosh(x)

Hyperbolic Secant

The reciprocal of hyperbolic cosine. Maximum value is 1 at x = 0.

sech(x) = 1 / cosh(x)

Hyperbolic Cosecant

The reciprocal of hyperbolic sine. Undefined at x = 0.

csch(x) = 1 / sinh(x)

Hyperbolic Cotangent

The reciprocal of hyperbolic tangent. Undefined at x = 0.

coth(x) = cosh(x) / sinh(x)

Key Properties of cosh(x)

Domain: All real numbers (-∞, +∞)
Range: [1, +∞) — cosh(x) is always greater than or equal to 1
Even function: cosh(-x) = cosh(x) — the graph is symmetric about the y-axis
Minimum value: cosh(0) = 1
Derivative: d/dx cosh(x) = sinh(x)
Integral: ∫ cosh(x) dx = sinh(x) + C
Identity: cosh²(x) - sinh²(x) = 1 (hyperbolic Pythagorean identity)

Relationship to Exponential Function

The hyperbolic cosine is intimately connected to the exponential function. In fact, it is the "even part" of e^x. Any exponential function can be decomposed into its even and odd parts: e^x = cosh(x) + sinh(x). This decomposition is analogous to how Euler's formula relates the regular cosine and sine to the complex exponential.

For complex arguments, the hyperbolic cosine is related to the regular cosine by: cosh(ix) = cos(x), establishing a deep connection between circular and hyperbolic functions through complex analysis.

Applications of Hyperbolic Cosine

Catenary curves: The shape of a freely hanging chain is described by y = a · cosh(x/a), making cosh essential in structural engineering.
Special relativity: The Lorentz factor and rapidity in special relativity involve hyperbolic functions. The relativistic velocity addition formula uses tanh.
Heat equation: Solutions to the heat equation on a finite interval involve cosh terms.
Electrical engineering: Transmission line equations use cosh and sinh to describe voltage and current distribution.
Machine learning: The softplus activation function log(1 + e^x) is related to cosh through various identities.

Common Values

cosh(0) = 1
cosh(1) = 1.5430806...
cosh(2) = 3.7621956...
cosh(3) = 10.067662...
cosh(ln 2) = 5/4 = 1.25

Tips for Working with Hyperbolic Functions

Hyperbolic functions mirror many trigonometric identities but with sign changes (Osborn's rule).
For large |x|, cosh(x) approximates e^|x|/2 since the smaller exponential term becomes negligible.
The Taylor series is cosh(x) = 1 + x²/2! + x⁴/4! + x⁶/6! + ... (even powers only).
Use the identity cosh²(x) - sinh²(x) = 1 to convert between hyperbolic functions.