Hyperbolic Functions Calculator

Calculate hyperbolic functions (sinh, cosh, tanh) and inverse hyperbolic functions. Visualize graphs and learn hyperbolic identities.

Hyperbolic Functions: Defined using exponential functions. For real number x:

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

Enter a real number to calculate hyperbolic functions
Range of x-values to display on the graph
x = 0
x = 0.5
x = 1
x = 1.5
x = 2
x = -1
x = π
x = ln(2)
Calculating...

Understanding Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric functions, but for a hyperbola rather than a circle. They appear in many areas of mathematics, including calculus, differential equations, and complex analysis.

Exponential Definitions:

sinh(x) = (eˣ - e⁻ˣ)/2

cosh(x) = (eˣ + e⁻ˣ)/2

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

Geometric Interpretation

1

Hyperbola: The point (cosh t, sinh t) lies on the right branch of the hyperbola x² - y² = 1, just as (cos t, sin t) lies on the circle x² + y² = 1.

2

Catenary: The shape of a hanging chain or cable is described by the hyperbolic cosine function: y = a cosh(x/a).

3

Area: The parameter t in (cosh t, sinh t) is twice the area of the hyperbolic sector defined by these points and the origin.

Applications of Hyperbolic Functions

  • Physics: Special relativity (Lorentz transformations), hanging cables (catenary)
  • Engineering: Transmission line theory, heat transfer, suspension bridge design
  • Mathematics: Solutions to differential equations, complex analysis, hyperbolic geometry
  • Electrical Engineering: Hyperbolic impedance, transmission lines
  • Architecture: Catenary arches (e.g., Gateway Arch in St. Louis)

Calculator Features:

  • Calculates all six hyperbolic functions (sinh, cosh, tanh, coth, sech, csch)
  • Calculates inverse hyperbolic functions (arsinh, arcosh, artanh, arcoth)
  • Verifies hyperbolic identities
  • Visualizes hyperbolic functions on interactive graphs
  • Compares hyperbolic and trigonometric functions

Frequently Asked Questions

sinh (hyperbolic sine) is defined using exponential functions: sinh(x) = (eˣ - e⁻ˣ)/2, while sin (circular sine) is defined using the unit circle. sinh(x) grows exponentially as x increases, while sin(x) oscillates between -1 and 1.

They are called hyperbolic functions because the points (cosh t, sinh t) form the right half of the equilateral hyperbola x² - y² = 1, analogous to how (cos t, sin t) form a circle x² + y² = 1.

  • arsinh(x): all real x
  • arcosh(x): x ≥ 1
  • artanh(x): -1 < x < 1
  • arcoth(x): |x| > 1
  • arsech(x): 0 < x ≤ 1
  • arcsch(x): x ≠ 0

Hyperbolic functions have simple derivatives and integrals: d/dx sinh(x) = cosh(x), d/dx cosh(x) = sinh(x). They appear in solutions to certain differential equations, particularly those modeling growth and decay processes, wave equations, and in integration techniques.

A catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. Its equation is y = a cosh(x/a), where a is a constant depending on the material and tension. The Gateway Arch in St. Louis is an inverted catenary.