Hyperbolic Functions Calculator — sinh, cosh, tanh and Inverses
Enter any real number x to instantly compute all six hyperbolic functions — sinh, cosh, tanh, coth, sech, csch — plus the three inverse functions arcsinh, arccosh, and arctanh. Select a function to plot its curve and highlight your x value.
Plot / highlight function
All six hyperbolic functions computed for the same x below
- 1
eˣ
e^1 = 2.718282 - 2
e⁻ˣ
e^(−1) = 0.367879 - 3
eˣ − e⁻ˣ
2.718282 − 0.367879 = 2.350402 - 4
sinh(x) = (eˣ − e⁻ˣ) ÷ 2
2.350402 ÷ 2 = 1.175201
How does this calculator work?
sinh x = (eˣ−e⁻ˣ)/2, cosh x = (eˣ+e⁻ˣ)/2, tanh = sinh/cosh, coth = 1/tanh, sech = 1/cosh, csch = 1/sinh. Inverses: arcsinh x = ln(x+√(x²+1)); arccosh x = ln(x+√(x²−1)) for x≥1; arctanh x = ½ln((1+x)/(1−x)) for |x|<1.
Formula
How this is calculated
Hyperbolic functions are analogues of trigonometric functions defined using the exponential function. sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2 are defined for all real x and satisfy cosh²x − sinh²x = 1 (the hyperbolic identity, analogous to cos²+sin²=1). The remaining four functions are reciprocals and quotients: tanh x = sinh/cosh, coth x = cosh/sinh, sech x = 1/cosh, csch x = 1/sinh. Note that coth and csch are undefined at x = 0.
The inverse hyperbolic functions express in terms of logarithms: arcsinh x = ln(x + √(x²+1)) (all reals), arccosh x = ln(x + √(x²−1)) (x ≥ 1 only), arctanh x = ½ ln((1+x)/(1−x)) (|x| < 1 only). These appear in integration formulas and in physics (special relativity, catenary curves, fluid mechanics).
The plot shows the selected function over x ∈ [−3, 3]. Coth and csch have a vertical asymptote at x = 0, so points with |y| > 20 are excluded to keep the chart readable.
Frequently asked questions
sin is defined by the unit circle (periodic, bounded −1 to 1). sinh is defined by the unit hyperbola: sinh x = (eˣ − e⁻ˣ)/2, which grows without bound. They share analogous identities but behave very differently for large x.
They appear in catenary curves (shape of a hanging chain), special relativity (rapidity), solutions to the wave equation, and the Laplace equation in cylindrical coordinates. Arccosh and arcsinh also arise in integration of √(x²±1) forms.
Because cosh x ≥ 1 for all real x (cosh 0 = 1, and cosh x grows from there). So the inverse can only receive values ≥ 1. Similarly, arctanh is only defined on (−1, 1) because |tanh x| < 1 for all real x.
TG we-Calculate Editorial Team. (2026). Hyperbolic Functions Calculator — sinh, cosh, tanh and Inverses [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/hyperbolic-functions-calculator
TG we-Calculate Editorial Team. "Hyperbolic Functions Calculator — sinh, cosh, tanh and Inverses." TG we-Calculate. 2026. https://we-calculate.com/calculator/hyperbolic-functions-calculator.
TG we-Calculate Editorial Team, "Hyperbolic Functions Calculator — sinh, cosh, tanh and Inverses," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/hyperbolic-functions-calculator
@misc{wecalculate_hyperbolic_functions_calculator, title = {Hyperbolic Functions Calculator — sinh, cosh, tanh and Inverses}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/hyperbolic-functions-calculator}}, year = {2026}, note = {TG we-Calculate} }
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