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Hyperbolic Functions

sinh(x)=(eˣ−e⁻ˣ)/2, cosh(x)=(eˣ+e⁻ˣ)/2, tanh(x)=sinh/cosh. Analogous to trig but for hyperbolas. Used in catenary cables, special relativity, and damped oscillations.

Concept Fundamentals
(eˣ−e⁻ˣ)/2
sinh
(eˣ+e⁻ˣ)/2
cosh
1
cosh²−sinh²
sinh/cosh
tanh

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cosh²x−sinh²x=1 (analog of cos²+sin²=1). Catenary: y=a·cosh(x/a) shapes hanging cables. tanh maps ℝ to (−1,1); used in neural networks.

Key quantities
(eˣ−e⁻ˣ)/2
sinh
Key relation
(eˣ+e⁻ˣ)/2
cosh
Key relation
1
cosh²−sinh²
Key relation
sinh/cosh
tanh
Key relation

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Why: Hyperbolic functions model catenary cables (cosh), relativistic velocity addition (tanh), and damped oscillations. They satisfy identities similar to trig but with sign differences.

How: sinh(x)=(eˣ−e⁻ˣ)/2, cosh(x)=(eˣ+e⁻ˣ)/2. tanh=sinh/cosh, coth=1/tanh, sech=1/cosh, csch=1/sinh. Identity: cosh²−sinh²=1.

cosh²x−sinh²x=1 (analog of cos²+sin²=1).Catenary: y=a·cosh(x/a) shapes hanging cables.
Sources:Khan Academy — Hyperbolic FunctionsWolfram MathWorld — Hyperbolic Sine

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Compute Hyperbolic Functionssinh, cosh, tanh, coth, sech, csch

Hyperbolic Functions Calculator

Enter Value

Results

sinh(x)
1.1752
cosh(x)
1.5431
tanh(x)
0.7616
coth(x)
1.3130
sech(x)
0.6481
csch(x)
0.8509

Calculation Steps

Input: x = 1.0000

Calculating basic hyperbolic functions:

sinh(x) = (e^x - e^(-x))/2 = (e^1.0000 - e^(-1.0000))/2 = 1.1752

cosh(x) = (e^x + e^(-x))/2 = (e^1.0000 + e^(-1.0000))/2 = 1.5431

tanh(x) = sinh(x)/cosh(x) = 1.1752/1.5431 = 0.7616

Calculating reciprocal hyperbolic functions:

coth(x) = 1/tanh(x) = 1/0.7616 = 1.3130

sech(x) = 1/cosh(x) = 1/1.5431 = 0.6481

csch(x) = 1/sinh(x) = 1/1.1752 = 0.8509

Graph of Hyperbolic Functions

-3-2-10123-4-3-2-101234xysinh(x)cosh(x)tanh(x)

What are Hyperbolic Functions?

Hyperbolic functions are analogs of the regular trigonometric functions but are defined in terms of exponential functions. They are named by analogy with the trigonometric functions, but they are defined using exponentials instead of angles.

The basic hyperbolic functions are:

  • Hyperbolic sine (sinh)
  • Hyperbolic cosine (cosh)
  • Hyperbolic tangent (tanh)
  • Hyperbolic cotangent (coth)
  • Hyperbolic secant (sech)
  • Hyperbolic cosecant (csch)

These functions have many applications in engineering, physics, mathematics, and other fields, especially for problems involving exponential growth and decay, oscillations, and wave propagation.

Properties of Hyperbolic Functions

  1. Definitions:

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

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

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

  2. Identities:

    cosh²(x) - sinh²(x) = 1

    1 - tanh²(x) = sech²(x)

    coth²(x) - 1 = csch²(x)

  3. Domain and Range:

    sinh(x) is defined for all real x, with range all real numbers

    cosh(x) is defined for all real x, with range [1, ∞)

    tanh(x) is defined for all real x, with range (-1, 1)

  4. Derivatives:

    d/dx[sinh(x)] = cosh(x)

    d/dx[cosh(x)] = sinh(x)

    d/dx[tanh(x)] = sech²(x)

Applications of Hyperbolic Functions

  • Catenary Curves:

    The shape of a hanging chain or cable is described by the hyperbolic cosine function.

  • Electric and Magnetic Fields:

    Hyperbolic functions are used in the analysis of electric and magnetic fields.

  • Signal Processing:

    Used in filter design and signal analysis.

  • Special Relativity:

    The Lorentz transformations can be expressed using hyperbolic functions.

  • Engineering:

    Used in the analysis of transmission lines, wave propagation, and structural engineering.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Catenary: y=cosh(x) — shape of hanging chain.

— Physics

⚛️

Relativity: v/c=tanh(rapidity).

— Special Relativity

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