Enter a value for x above and click Calculate to see all 12 hyperbolic function values.
What Are Hyperbolic Functions?
Hyperbolic functions are analogues of the trigonometric functions, but defined using the hyperbola x² − y² = 1 rather than the unit circle x² + y² = 1. Just as (cos θ, sin θ) traces the unit circle, the pair (cosh t, sinh t) traces the right branch of the unit hyperbola.
The six hyperbolic functions — sinh, cosh, tanh, csch, sech, and coth — appear across engineering, physics, and applied mathematics. They describe the shapes of hanging cables, the geometry of special relativity, solutions to many differential equations, and signal behaviour in electrical engineering.
Unlike trigonometric functions, which are periodic, hyperbolic functions grow without bound as x increases. The deep link between the two families is Euler's formula extended to imaginary arguments: sinh(x) = −i sin(ix) and cosh(x) = cos(ix), where i = √(−1).
Shape of the Three Primary Hyperbolic Functions
The diagram below shows sinh (teal S-curve), cosh (blue U-curve), and tanh (purple bounded S-curve) over x ∈ [−3, 3]. Dashed lines mark the asymptotes y = ±1 that tanh never crosses.
Exponential decomposition
eˣ = cosh(x) + sinh(x)
cosh is the even part of eˣ, sinh is the odd part. Every eˣ splits into these two halves.
The catenary
y = a · cosh(x / a)
The shape a hanging chain takes under gravity — the cosh U-curve scaled by a constant a.
The Six Hyperbolic Functions — Definitions and Formulas
Every hyperbolic function is expressed in terms of the exponential function eˣ. The two primary functions are sinh (hyperbolic sine) and cosh (hyperbolic cosine); the remaining four are derived from these two.
| Function | Definition (exponential form) | Domain | Range |
|---|---|---|---|
| sinh(x) | (eˣ − e⁻ˣ) / 2 | (−∞, ∞) | (−∞, ∞) |
| cosh(x) | (eˣ + e⁻ˣ) / 2 | (−∞, ∞) | [1, ∞) |
| tanh(x) | sinh(x) / cosh(x) | (−∞, ∞) | (−1, 1) |
| csch(x) | 1 / sinh(x) | x ≠ 0 | (−∞,0) ∪ (0,∞) |
| sech(x) | 1 / cosh(x) | (−∞, ∞) | (0, 1] |
| coth(x) | cosh(x) / sinh(x) | x ≠ 0 | (−∞,−1) ∪ (1,∞) |
Key observations
sinh is an odd function: sinh(−x) = −sinh(x). Its graph passes through the origin and is anti-symmetric.
cosh is an even function: cosh(−x) = cosh(x). It has a minimum of 1 at x = 0 — it can never be zero or negative.
tanh is bounded: tanh always lies strictly between −1 and 1, approaching ±1 asymptotically. This makes it a popular activation function in neural networks.
Inverse Hyperbolic Functions — Definitions and Domains
The inverse hyperbolic functions undo their forward counterparts. Because hyperbolic functions are not all one-to-one over all of ℝ, some inverses have restricted domains. Each inverse has a closed-form expression involving the natural logarithm.
| Function | Logarithmic form | Domain | Range |
|---|---|---|---|
| arcsinh(x) | ln(x + √(x²+1)) | (−∞, ∞) | (−∞, ∞) |
| arccosh(x) | ln(x + √(x²−1)) | x ≥ 1 | [0, ∞) |
| arctanh(x) | ½ ln((1+x)/(1−x)) | |x| < 1 | (−∞, ∞) |
| arccsch(x) | arcsinh(1/x) | x ≠ 0 | x ≠ 0 |
| arcsech(x) | arccosh(1/x) | 0 < x ≤ 1 | [0, ∞) |
| arccoth(x) | arctanh(1/x) | |x| > 1 | x ≠ 0 |
Why domains are restricted
The inverse of a function only exists where the original function is one-to-one. arccosh is only defined for x ≥ 1 because cosh is only one-to-one on that half (its minimum is at x = 0, so the function is symmetric and not invertible across its full domain). Similarly, arctanh requires |x| < 1 because tanh maps (−∞, ∞) onto (−1, 1), so only values in that open interval have a well-defined pre-image.
Hyperbolic Identities You Should Know
Hyperbolic identities mirror trigonometric identities in form, but with crucial sign differences. The most important is the Pythagorean identity for hyperbolics — a minus sign where circular trigonometry has a plus sign.
| Category | Identity |
|---|---|
| Pythagorean | cosh²(x) − sinh²(x) = 1 |
| Pythagorean | 1 − tanh²(x) = sech²(x) |
| Double angle | sinh(2x) = 2 sinh(x) cosh(x) |
| Double angle | cosh(2x) = cosh²(x) + sinh²(x) |
| Reciprocal | csch(x) = 1 / sinh(x) |
| Reciprocal | sech(x) = 1 / cosh(x) • coth(x) = 1 / tanh(x) |
| Derivative | d/dx sinh(x) = cosh(x) |
| Derivative | d/dx cosh(x) = sinh(x) |
| Derivative | d/dx tanh(x) = sech²(x) |
Osborne's Rule
To convert any trigonometric identity to its hyperbolic equivalent, replace sin → sinh and cos → cosh, then negate every product or implied product of two sinh functions. This is why cos²θ + sin²θ = 1 becomes cosh²x − sinh²x = 1 (the product sinh·sinh picks up a minus sign).
Where Hyperbolic Functions Appear in Engineering and Physics
The Catenary — Hanging Cables
A flexible chain or rope hanging under gravity takes the shape y = a cosh(x/a). This catenary curve appears in suspension bridges, power lines, and the Gateway Arch in St Louis, which was deliberately designed as an inverted catenary.
Special Relativity — Rapidity
In special relativity, the rapidity φ of a moving object is related to velocity v and speed of light c by tanh(φ) = v/c. Rapidities add linearly for collinear boosts, unlike velocities, which is why arctanh(v/c) appears so naturally in relativistic mechanics.
Differential Equations
The general solution to y'' − y = 0 (which arises in heat transfer, wave propagation, and beam deflection) is y = A cosh(x) + B sinh(x). Hyperbolic functions are as fundamental to this equation as sin/cos are to y'' + y = 0.
Electrical Engineering
Transmission line equations involve hyperbolic functions. The voltage and current along a lossy transmission line are expressed using cosh and sinh of the propagation constant times distance. The Langevin function L(x) = coth(x) − 1/x models paramagnetism.
Machine Learning — Activations
tanh is one of the original neural network activation functions. It maps any real input to (−1, 1), centres its output near zero, and has a clean derivative: tanh'(x) = 1 − tanh²(x) = sech²(x). It was widely used in early recurrent neural networks.
Hyperbolic Geometry
In the Poincaré disk model of hyperbolic geometry, distances and angles are computed using inverse hyperbolic functions. Hyperbolic space has constant negative curvature, and its metric naturally involves arcsinh and arctanh, making this calculator directly useful for studying non-Euclidean geometry.
Hyperbolic Functions vs Trigonometric Functions
The parallel between hyperbolic and trigonometric functions is deep but imperfect. Both families satisfy Pythagorean-style identities, both have double-angle formulas, and both arise as solutions to simple second-order differential equations. The differences lie in the sign of one term and the nature of the geometry.
| Property | Trigonometric | Hyperbolic |
|---|---|---|
| Pythagorean identity | cos²x + sin²x = 1 | cosh²x − sinh²x = 1 |
| Period | 2π (periodic) | None (non-periodic) |
| Range of cos/cosh | [−1, 1] | [1, ∞) |
| Geometry | Unit circle x²+y²=1 | Unit hyperbola x²−y²=1 |
| d/dx of cos/cosh | −sin(x) | +sinh(x) |
| Exponential link | cos x = Re(eⁱˣ) | cosh x = (eˣ + e⁻ˣ)/2 |
Frequently Asked Questions
What is the difference between sinh and sin?
Why is cosh(x) always ≥ 1?
What is the domain of arccosh(x)?
How are hyperbolic functions related to the exponential function?
Where do hyperbolic functions appear in real life?
Why does csch(0) not exist?
What is the derivative of tanh(x)?
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