Hyperbolic Functions Calculator

Calculate hyperbolic and inverse hyperbolic functions: sinh, cosh, tanh, and more

Calculate Hyperbolic Functions

Domain: All real numbers

Function Values

x =0
sinh(x) =0
cosh(x) =1.000000
tanh(x) =0
coth(x) =
sech(x) =1.000000
csch(x) =

Formulas Used:

sinh(x) = (e^x - e^(-x)) / 2

cosh(x) = (e^x + e^(-x)) / 2

tanh(x) = sinh(x) / cosh(x)

Hyperbolic Function Formulas

Forward Functions

sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
coth(x) = cosh(x) / sinh(x)
sech(x) = 1 / cosh(x)
csch(x) = 1 / sinh(x)

Inverse Functions

arsinh(x) = ln(x + √(x² + 1))
arcosh(x) = ln(x + √(x² - 1))
artanh(x) = ½ln((1+x)/(1-x))
arcoth(x) = ½ln((x+1)/(x-1))
arsech(x) = ln((1+√(1-x²))/x)
arcsch(x) = ln(1/x + √(1/x² + 1))

Common Values

At x = 0

sinh(0) = 0

cosh(0) = 1

tanh(0) = 0

At x = 1

sinh(1) ≈ 1.175

cosh(1) ≈ 1.543

tanh(1) ≈ 0.762

At x = ln(2)

sinh(ln(2)) = 0.75

cosh(ln(2)) = 1.25

tanh(ln(2)) = 0.6

Function Properties

Parity

sinh, tanh: odd functions

cosh, sech: even functions

Range

sinh: (-∞, ∞)

cosh: [1, ∞)

tanh: (-1, 1)

Identities

cosh²(x) - sinh²(x) = 1

1 - tanh²(x) = sech²(x)

Quick Tips

Hyperbolic functions relate to hyperbolas like trig functions to circles

Unlike trig functions, hyperbolic functions are not periodic

cosh(x) ≥ 1 for all real x

tanh(x) approaches ±1 as x approaches ±∞

Understanding Hyperbolic Functions

What are Hyperbolic Functions?

Hyperbolic functions are analogues of trigonometric functions but are based on hyperbolas rather than circles. They are defined using exponential functions and have many similar properties to trigonometric functions.

Key Differences from Trigonometric Functions

  • Not periodic (don't repeat values)
  • Based on exponential functions rather than circular motion
  • Points (cosh x, sinh x) form a hyperbola, not a circle

Applications

Physics

Special relativity, wave equations, heat transfer

Engineering

Catenary curves (hanging cables), structural analysis

Mathematics

Complex analysis, differential equations, calculus

Computer Science

Machine learning, neural networks (activation functions)