Hyperbolic Functions Calculator

The odd and even halves of the exponential

Split eˣ into a part that is symmetric about zero and a part that is antisymmetric. The symmetric half is cosh, the antisymmetric half is sinh, and adding them back gives the exponential you started with.

The tool shows eˣ and e⁻ˣ on their own line for that reason. At x = 1 they are 2.718281828 and 0.3678794412, and cosh x + sinh x = eˣ — 2.718281828 against 2.718281828, agreeing.

Every other identity is bookkeeping on those two numbers. The double-angle formula, the addition formulae, the derivative relationships — none of them needs to be memorised separately once you see them as statements about eˣ and e⁻ˣ.

The minus sign, and where the name comes from

cosh²x − sinh²x came to 1 from the values above. It is 1 for every x, and that MINUS sign is the whole difference from cos²+sin²: the point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, which is where the name comes from.

Trace the point (cosh t, sinh t) as t runs over the reals and you get the right branch of the hyperbola x² − y² = 1 — which is exactly why these are the HYPERBOLIC functions, and why the circular ones, which trace x² + y² = 1, are called circular.

cosh²x − sinh²x came to 1.00000003 from the values above. It is 1 for every x, and that MINUS sign is the whole difference from cos²+sin²: the point (cosh t, sinh t) lies on the hyperbola x² − y² = 1, which is where the name comes from. The digits after the 1 are floating-point cancellation, not an error in the identity: at x = 10 both squares are about 121291299.4, and subtracting two numbers that size to get 1 throws away roughly 8 significant figures.

The parameter t is worth being careful about. For the circular functions it is an angle; here it is twice the AREA of the sector swept out. That is why the inverses are written arsinh, arcosh and artanh — ar for area — and not arcsinh.

Not periodic, and no amount of reduction will help

sin and cos repeat every 2π, so any argument can be reduced to a standard range. sinh and cosh do not repeat at all: they grow like eˣ/2 forever.

At x = 3, sinh is already 10.01787493 and cosh is 10.067662. At x = 10 they are 11013.23287 and 11013.23292 — nearly equal, because e⁻ˣ has become negligible beside eˣ.

At x = 3, e⁻ˣ is 0.04978706837 against eˣ at 20.08553692. Both terms still matter at this size, which is why sinh and cosh are noticeably different here and separate as x grows.

Two useful consequences: cosh x ≥ 1 for every x, with equality only at zero — so arcosh needs x ≥ 1, which is a domain restriction rather than an arbitrary rule. And sinh is one-to-one on the whole line, so arsinh is defined everywhere, unlike arcsin.

tanh is the bounded one

tanh x = sinh x / cosh x is always strictly between −1 and 1, approaching them without ever arriving. At x = 3 it is 0.9950547537; at x = 10 it is 0.9999999959.

That boundedness, combined with being smooth, monotonic and having a simple derivative (1 − tanh²), is why tanh was the standard squashing function in neural networks for years, and why it appears in every physics problem where a quantity saturates.

coth is the same ratio inverted, which puts it outside (−1, 1) instead of inside, and gives it a vertical asymptote at zero. coth x = cosh x / sinh x, and sinh 0 = 0 — so coth is undefined at x = 0, with a vertical asymptote there.

The inverses have closed forms, unlike the circular ones

arcsin has no expression in elementary functions. arsinh does: arsinh x = ln(x + √(x² + 1)). All six inverses are logarithms, and the tool gives each with its domain.

The reason is that the hyperbolic functions are built from exponentials, so inverting them means inverting an exponential — which is a logarithm. The circular functions are not, and there is nothing to invert into.

The domains are worth reading rather than memorising, because each has a reason. At x = 0.5: asinh = 0.4812118251, atanh = 0.5493061443, asech = 1.316957897, acsch = 1.443635475, while acosh and acoth are outside their domains there. At x = 2 the situation reverses — x = 2 is outside the domain of atanh: −1 < x < 1.

Where they turn up

A hanging chain traces a catenary, y = a·cosh(x/a) — not a parabola, which is the shape people usually guess. The distinction is real: the arch in St Louis is an inverted weighted catenary, and a parabola would not have stood.

They also run special relativity, where a Lorentz boost is a rotation through an imaginary angle and rapidity is the hyperbolic analogue of an angle. Velocities do not add; rapidities do, exactly as angles add for ordinary rotations, and the addition formula for tanh is the relativistic velocity-addition law.

And they solve the differential equation y″ = y, which is what makes them the natural basis for problems with exponential growth in both directions — cables, heat, and the shape of a soap film between two rings.

sinh is odd and cosh is even, exactly like sin and cos — which is one of the places the analogy does hold cleanly.

Sources and methodology

The hyperbolic functions and their identities are standard; the reference below carries the canonical definitions.

• NIST Digital Library of Mathematical Functions — §4.28, Hyperbolic Functions
• NIST DLMF — §4.37, Inverse Hyperbolic Functions
• How we calculate — methodology

Method. Every figure on this page comes from src/lib/hyperbolic-functions.ts over src/lib/algebra/special.ts. tanh is computed through expm1 rather than as a ratio of exponentials, so it stays accurate at large |x| where both would overflow. The identity checks are evaluated from the returned values rather than recomputed, so they test what the page actually displays. That engine is verified on every change against 76 hand-written assertions, including that cosh²−sinh² returns 1 to within the cancellation the magnitude allows, and that each inverse round-trips. The count and the per-case breakdown are published on the formula verification page.

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Read the guide

The hyperbolic functions solve y″ = y, which is the same family of problems the Doubling Time Calculator handles from the growth-rate side. For multi-step expressions involving them, the Scientific Calculator evaluates whole formulas at once.