Hyperbolic Function Calculators

Professional online calculators for hyperbolic and inverse hyperbolic functions

Hyperbolic Functions

sinh

Hyperbolic Sine (sinh x)

Hyperbolic sine - fundamental function of hyperbolic geometry

cosh

Hyperbolic Cosine (cosh x)

Hyperbolic cosine - describes catenary curves and exponential functions

tanh

Hyperbolic Tangent (tanh x)

Hyperbolic tangent - important in neurobiology and physics

coth

Hyperbolic Cotangent (coth x)

Hyperbolic cotangent - reciprocal of hyperbolic tangent

sech

Hyperbolic Secant (sech x)

Hyperbolic secant - reciprocal of hyperbolic cosine

csch

Hyperbolic Cosecant (csch x)

Hyperbolic cosecant - reciprocal of hyperbolic sine

Inverse Hyperbolic Functions

asinh

Inverse Hyperbolic Sine (asinh x)

Area hyperbolic sine - inverse function of hyperbolic sine

acosh

Inverse Hyperbolic Cosine
(acosh x)

Area hyperbolic cosine - inverse function of hyperbolic cosine

atanh

Inverse Hyperbolic Tangent (atanh x)

Area hyperbolic tangent - inverse function of hyperbolic tangent

acoth

Inverse Hyperbolic Cotangent (acoth x)

Area hyperbolic cotangent - inverse function of hyperbolic cotangent

asech

Inverse Hyperbolic Secant
(asech x)

Area hyperbolic secant - inverse function of hyperbolic secant

acsch

Inverse Hyperbolic Cosecant (acsch x)

Area hyperbolic cosecant - inverse function of hyperbolic cosecant

About Hyperbolic Functions

Hyperbolic functions are fundamental mathematical functions of great importance in various fields of science and engineering:

Physics - Relativity theory
Engineering - Catenary curves
Mathematics - Hyperbolic geometry

Biology - Growth models
Statistics - Distributions
AI/ML - Activation functions

Important Function Classes

Basic Functions

sinh x = (eˣ - e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
tanh x = sinh x / cosh x

Reciprocals

coth x = 1/tanh x
sech x = 1/cosh x
csch x = 1/sinh x

Identities

cosh² x - sinh² x = 1
tanh² x + sech² x = 1
coth² x - csch² x = 1

Inverse Functions

asinh x = ln(x + √(x² + 1))
acosh x = ln(x + √(x² - 1))
atanh x = ½ln((1+x)/(1-x))

Tip: Hyperbolic functions have similar properties to trigonometric functions, but are based on the exponential function rather than the unit circle.

Quick Reference

sinh 0 = 0

Hyperbolic sine

cosh 0 = 1

Hyperbolic cosine

tanh 0 = 0

Hyperbolic tangent

sech 0 = 1

Hyperbolic secant

cosh² x - sinh² x = 1

Main identity

Function Properties

Domain & Range:

sinh, cosh

ℝ → ℝ

tanh

ℝ → (-1, 1)

atanh

(-1, 1) → ℝ

acosh

[1, ∞) → [0, ∞)