Hyperbolic Functions Calculator
Calculate hyperbolic sine, cosine, tangent, and their inverses.
Quick Examples
Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions, defined using the hyperbola rather than the circle.
Definitions
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
Key Properties
| Function | Domain | Range |
|---|---|---|
| sinh(x) | All real | All real |
| cosh(x) | All real | [1, ∞) |
| tanh(x) | All real | (-1, 1) |
| asinh(x) | All real | All real |
| acosh(x) | [1, ∞) | [0, ∞) |
| atanh(x) | (-1, 1) | All real |
Identities
This is the hyperbolic analog of the Pythagorean identity.
Applications
- Catenary curve (sagging cables)
- Special relativity
- Electrical engineering
- Complex number trigonometry
A Short History
Hyperbolic functions were introduced in the 1760s by Vincenzo Riccati and Johann Heinrich Lambert, with Leonhard Euler developing the modern treatment. Euler showed that (cosh t, sinh t) parametrizes the hyperbola x² − y² = 1 exactly the way (cos t, sin t) parametrizes the unit circle — the source of the name.
Worked Example: Computing by Hand
Evaluate sinh(1), cosh(1), and tanh(1) from the definitions:
- Compute the exponentials: e¹ ≈ 2.718281828, e⁻¹ ≈ 0.367879441.
- sinh(1) = (2.718281828 − 0.367879441)/2 ≈ 1.175201194.
- cosh(1) = (2.718281828 + 0.367879441)/2 ≈ 1.543080635.
- tanh(1) = 1.175201194 / 1.543080635 ≈ 0.761594156.
- Check with the identity: 1.543080635² − 1.175201194² ≈ 1.000000000.
All Six Functions at a Glance
| Function | Definition | Domain | Range | Parity |
|---|---|---|---|---|
| sinh x | (eˣ − e⁻ˣ)/2 | All real | All real | Odd |
| cosh x | (eˣ + e⁻ˣ)/2 | All real | [1, ∞) | Even |
| tanh x | sinh x / cosh x | All real | (−1, 1) | Odd |
| coth x | cosh x / sinh x | x ≠ 0 | (−∞, −1) ∪ (1, ∞) | Odd |
| sech x | 1 / cosh x | All real | (0, 1] | Even |
| csch x | 1 / sinh x | x ≠ 0 | All real except 0 | Odd |
Derivatives and Integrals
Hyperbolic derivatives carry no sign traps like the trig versions — d/dx[cosh x] = +sinh x:
| Function | Derivative | Indefinite integral |
|---|---|---|
| sinh x | cosh x | cosh x + C |
| cosh x | sinh x | sinh x + C |
| tanh x | sech²x | ln(cosh x) + C |
| coth x | −csch²x | ln|sinh x| + C |
| sech x | −sech x·tanh x | arctan(sinh x) + C |
| csch x | −csch x·coth x | ln|tanh(x/2)| + C |
Addition and Double-Angle Identities
cosh(x ± y) = cosh x·cosh y ± sinh x·sinh y
sinh(2x) = 2·sinh x·cosh x
cosh(2x) = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x
sech²x + tanh²x = 1
A handy conversion trick (Osborn's rule): take any trigonometric identity, replace sin with sinh and cos with cosh, and flip the sign of any term containing a product of two sines. So cos²x + sin²x = 1 becomes cosh²x − sinh²x = 1, and cos(x+y) = cos x cos y − sin x sin y becomes cosh(x+y) = cosh x cosh y + sinh x sinh y.
Hyperbolic vs Circular Functions
| Property | Circular (sin, cos, tan) | Hyperbolic (sinh, cosh, tanh) |
|---|---|---|
| Underlying curve | Unit circle x² + y² = 1 | Hyperbola x² − y² = 1 |
| Key identity | sin²x + cos²x = 1 | cosh²x − sinh²x = 1 |
| Behavior for large x | Bounded oscillation | Exponential growth |
| Periodicity | 2π-periodic | Not periodic |
| Argument units | Degrees or radians | Dimensionless pure number |
Inverse Hyperbolic Functions: The Logarithm Formulas
acosh(x) = ln(x + √(x² − 1)), x ≥ 1
atanh(x) = ½·ln((1 + x)/(1 − x)), |x| < 1
The domain restrictions come straight from the ranges above: acosh(0.5) is undefined because cosh never dips below 1, and atanh(1) is undefined because tanh never quite reaches 1. Example: asinh(1) = ln(1 + √2) ≈ 0.881374.
Case Study: Sag in a Power Line
A transmission line spans 200 m between two equal-height towers, with a catenary parameter a = 800 m. The shape measured from the lowest point is y(x) = 800·cosh(x/800). At the tower, x = 100 m, so x/a = 0.125 and cosh(0.125) ≈ 1.0078227. The sag is 800 × (1.0078227 − 1) ≈ 6.26 m. A simple parabola approximation gives 100²/(2·800) = 6.25 m — close for a taut span, but the exact catenary matters for longer spans.
Common Misconceptions
- The identity has a minus sign: cosh²x − sinh²x = 1, not plus. The sign flip breaks every identity derived from it.
- tanh never reaches 1: tanh(5) ≈ 0.9999092 and tanh(10) ≈ 0.9999999959, but it is always strictly below 1 — the line y = 1 is an asymptote.
- Not periodic: hyperbolic functions grow monotonically on each side of zero; adding a constant never returns the same value.
- The argument is not an angle: it is a dimensionless number (geometrically, twice the area of a hyperbolic sector).
Series Expansions
Near zero, hyperbolic functions are polynomials to any desired accuracy:
cosh(x) = 1 + x²/2! + x⁴/4! + …
For small x, sinh(x) ≈ x and cosh(x) ≈ 1 + x²/2 — the first term of each series is a useful quick estimate.
Solving Hyperbolic Equations
Solve cosh(x) = 5/4. Write it in exponentials: (eˣ + e⁻ˣ)/2 = 5/4, so eˣ + e⁻ˣ = 5/2. Multiply through by eˣ: (eˣ)² − (5/2)eˣ + 1 = 0, a quadratic in eˣ with solutions eˣ = 2 or eˣ = 1/2. Hence x = ln(2) or x = −ln(2). Check: cosh(ln 2) = (2 + 1/2)/2 = 5/4, confirming both solutions.
Key Values
| x | sinh(x) | cosh(x) | tanh(x) |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 1 | 1.1752 | 1.5431 | 0.7616 |
| 2 | 3.6269 | 3.7622 | 0.9640 |
| −1 | −1.1752 | 1.5431 | −0.7616 |
The odd/even symmetry is visible directly: sinh and tanh flip sign with x, cosh does not.
The Complex Connection
Hyperbolic and circular functions are two faces of the same exponential coin:
cosh(ix) = cos(x)
Feed an imaginary argument to a hyperbolic function and ordinary trigonometry comes out. This is why identities translate so cleanly between the two families (via Osborn's rule above) and why hyperbolic functions appear in solutions to differential equations with real exponentials.
tanh in Machine Learning
The hyperbolic tangent is a classic neural-network activation function: it squashes any input into (−1, 1), is centered at zero, and has a simple derivative (sech²x = 1 − tanh²x) that makes gradient-based training efficient. Its S-shaped curve is the zero-centered cousin of the logistic sigmoid.
Graph Shapes at a Glance
- sinh: an S-curve through the origin, steeper than a line, growing exponentially in both directions (odd symmetry).
- cosh: a U-shape with minimum value 1 at x = 0 — the catenary curve (even symmetry).
- tanh: an S-curve flattening toward ±1, the classic squashing function (odd symmetry).
Because sinh and tanh are odd and cosh is even, identities involving sign changes simplify immediately: sinh(−x) = −sinh(x), cosh(−x) = cosh(x).