Definitions
cosh x = (eˣ + e⁻ˣ) ÷ 2
tanh x = sinh x ÷ cosh x = (eˣ − e⁻ˣ) ÷ (eˣ + e⁻ˣ)
coth x = 1 ÷ tanh x · sech x = 1 ÷ cosh x · csch x = 1 ÷ sinh x
At x = 0: sinh 0 = 0, cosh 0 = 1 and tanh 0 = 0. cosh x is never below 1, and tanh x always lies between −1 and 1.
Why "hyperbolic"?
For every x, the point (cosh x, sinh x) lies on the hyperbola X² − Y² = 1, just as (cos t, sin t) lies on the circle X² + Y² = 1. That gives the key identity:
A hanging chain or cable takes the shape of a cosh curve, called a catenary, and tanh appears in physics and as an activation function in neural networks.
Derivatives
{_tbl(["Function", "Derivative", "Compare with"], [ ["sinh x", "cosh x", "(sin x)′ = cos x"], ["cosh x", "sinh x", "(cos x)′ = −sin x: no minus sign here"], ["tanh x", "sech² x = 1 − tanh² x", "(tan x)′ = sec² x"], ["coth x", "−csch² x", ""], ["sech x", "−sech x · tanh x", ""], ["csch x", "−csch x · coth x", ""]])}These follow straight from the definitions: the derivative of (eˣ − e⁻ˣ) ÷ 2 is (eˣ + e⁻ˣ) ÷ 2, so (sinh x)′ = cosh x.
Integrals
Inverse hyperbolic functions
{_tbl(["Inverse", "As a logarithm", "Domain"], [ ["arsinh x", "ln(x + √(x² + 1))", "all real x"], ["arcosh x", "ln(x + √(x² − 1))", "x ≥ 1"], ["artanh x", "½ ln((1 + x) ÷ (1 − x))", "−1 < x < 1"]])}Work them out
The hyperbolic functions calculator gives sinh, cosh, tanh, their reciprocals and inverses for any value, with the formula shown. On a scientific calculator, look for a hyp key: press it, then sin, cos or tan.
Every worked example on this page was checked with Calcist's own engine and an independent reference.