Hyperbolic Functions Calculator (sinh, cosh, tanh & inverses)

Q: Why is it called "hyperbolic"?

Because (cosh t, sinh t) lies on the unit hyperbola x² − y² = 1, just as (cos t, sin t) lies on the unit circle.

Q: How are hyperbolic and trigonometric functions related?

Through complex numbers: cosh x = cos(ix) and sinh x = −i sin(ix).

Q: What curve does a hanging chain form?

A catenary, y = a cosh(x ÷ a), where a depends on the horizontal tension and weight per unit length.

Understanding the hyperbolic functions calculator — concept and principle

Hyperbolic functions are combinations of the exponential functions eˣ and e⁻ˣ that behave like trigonometric functions for a hyperbola instead of a circle. This calculator returns sinh, cosh and tanh of x and the inverse functions asinh, acosh and atanh. They appear in the shape of a hanging cable (the catenary), in special relativity, in heat-transfer and transmission-line equations, in solutions of differential equations, and as the tanh activation function in neural networks.

cosh is the average of eˣ and e⁻ˣ, sinh is half their gap — and a hanging chain takes the cosh shape.

How it is calculated

sinh x = (eˣ − e⁻ˣ) ÷ 2, cosh x = (eˣ + e⁻ˣ) ÷ 2, tanh x = sinh x ÷ cosh x.
asinh x = ln(x + √(x² + 1)), defined for all x.
acosh x = ln(x + √(x² − 1)), defined for x ≥ 1 (non-negative branch).
atanh x = ½ ln((1 + x) ÷ (1 − x)), defined for −1 < x < 1.

x is a pure number, not an angle in degrees.

The principle

For a circle x² + y² = 1 the point (cos t, sin t) traces the curve. For the hyperbola x² − y² = 1 the point (cosh t, sinh t) does the same, which is why cosh²x − sinh²x = 1, the hyperbolic counterpart of sin² + cos² = 1. The parameter t equals twice the area of the hyperbolic sector, just as an angle in radians is twice the area of a circular sector of radius 1. The inverse formulas come from solving y = sinh x for x: writing u = eˣ gives a quadratic u² − 2yu − 1 = 0, whose positive root leads to the logarithm form.

Worked example

Default input: x = 1, and e ≈ 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.
asinh 1 = ln(1 + √2) = ln 2.414213562 = 0.881373587.
acosh 1 = ln(1 + 0) = 0; atanh 1 = ½ ln(2 ÷ 0) is infinite, so the calculator shows —.

Each point is the calculator re-run with only “x” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

x	sinh
0.25	0.2526123168
0.5	0.5210953055
0.75	0.8223167319
1	1.175201194
1.25	1.60191908
1.5	2.129279455
2	3.626860408

How to read the result

cosh x is always at least 1 and is even (same for x and −x); sinh and tanh are odd. tanh always lies between −1 and 1 and approaches ±1 quickly: tanh 3 ≈ 0.995. For large x, sinh x and cosh x are both close to eˣ ÷ 2, so they grow exponentially. For small x, sinh x ≈ x and tanh x ≈ x.

Common mistakes

Treating x as an angle in degrees; hyperbolic functions take a real number.
Entering x < 1 for acosh or |x| ≥ 1 for atanh, which have no real result.
Expecting periodic behaviour as with sin and cos; real hyperbolic functions are not periodic.

Keep in mind: For large |x| (above about 710), eˣ overflows double-precision arithmetic and sinh and cosh return infinity. The results are numerical approximations to 10 significant digits.

Frequently asked questions

Why is it called "hyperbolic"?