Hyperbolic Functions Calculator
Evaluate sinh, cosh, tanh and their reciprocals coth, sech and csch for any real number, from the exponential definitions.
How it works
sinh x = (eˣ − e⁻ˣ)/2 · cosh x = (eˣ + e⁻ˣ)/2 · tanh x = sinh x / cosh x
The hyperbolic functions are built from the exponential eˣ rather than from a circle. sinh is the odd combination (eˣ − e⁻ˣ)/2 and cosh is the even combination (eˣ + e⁻ˣ)/2; their ratio is tanh. They satisfy cosh²x − sinh²x = 1, mirroring the circular identity with a minus sign, and describe shapes like the hanging catenary of a chain or cable. cosh is always at least 1, so sech is always defined, but coth and csch are undefined at x = 0 where sinh is zero.
Worked example
At x = 1: sinh 1 = (e − 1/e)/2 = (2.7183 − 0.3679)/2 = 1.1752, cosh 1 = (2.7183 + 0.3679)/2 = 1.5431, and tanh 1 = 1.1752 ÷ 1.5431 = 0.7616.
Sources: Hyperbolic functions — Wikipedia
Frequently asked questions
How do hyperbolic functions differ from ordinary trig functions?
Trigonometric functions parametrise a circle (x² + y² = 1) and are periodic; hyperbolic functions parametrise a hyperbola (x² − y² = 1) using the exponential and are not periodic. The argument x is a real number, not an angle.
Where do hyperbolic functions appear in real life?
A flexible cable or chain hanging under gravity forms a catenary, whose curve is described by cosh. They also arise in special relativity, heat flow and the shape of some arches.
Why are coth and csch undefined at zero?
Both are reciprocals of sinh, and sinh 0 = 0. Dividing by zero is undefined, so the calculator reports "undefined" for coth and csch when x = 0. cosh 0 = 1, so sech stays defined.
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This calculator is for educational purposes. Double-check important results before acting on them.