Hyperbolic Functions Calculator

Calculate hyperbolic functions (sinh, cosh, tanh, csch, sech, coth) and their inverses

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Understanding Hyperbolic Functions

Hyperbolic functions are analogous to the circular trigonometric functions but are defined in terms of the exponential function. They appear in various areas of mathematics, physics, and engineering.

Basic Definitions

Hyperbolic Sine (sinh)

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

Hyperbolic Cosine (cosh)

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

Hyperbolic Tangent (tanh)

tanh(x) = sinh(x) / cosh(x) = (e^x - e^-x) / (e^x + e^-x)

Hyperbolic Cosecant (csch)

csch(x) = 1 / sinh(x) = 2 / (e^x - e^-x)

Hyperbolic Secant (sech)

sech(x) = 1 / cosh(x) = 2 / (e^x + e^-x)

Hyperbolic Cotangent (coth)

coth(x) = cosh(x) / sinh(x) = (e^x + e^-x) / (e^x - e^-x)

Inverse Hyperbolic Functions

Inverse Hyperbolic Sine (asinh)

asinh(x) = ln(x + √(x² + 1))

Inverse Hyperbolic Cosine (acosh)

acosh(x) = ln(x + √(x² - 1)), for x ≥ 1

Inverse Hyperbolic Tangent (atanh)

atanh(x) = 0.5 * ln((1 + x) / (1 - x)), for |x| < 1

Inverse Hyperbolic Cosecant (acsch)

acsch(x) = ln(1/x + √(1/x² + 1)), for x ≠ 0

Inverse Hyperbolic Secant (asech)

asech(x) = ln((1 + √(1-x²)) / x), for 0 < x ≤ 1

Inverse Hyperbolic Cotangent (acoth)

acoth(x) = 0.5 * ln((x + 1) / (x - 1)), for |x| > 1

Key Properties

sinh(-x) = -sinh(x) (odd function)
cosh(-x) = cosh(x) (even function)
cosh²(x) - sinh²(x) = 1
1 - tanh²(x) = sech²(x)
coth²(x) - 1 = csch²(x)

Applications

Hyperbolic functions appear in many areas including:

Solutions to differential equations
Calculations of the shapes of hanging cables (catenaries)
Electric and magnetic fields
Signal processing
Special relativity in physics
Geometric calculations involving hyperbolas

Examples

Example 1: sinh(1) = (e¹ - e⁻¹) / 2 ≈ 1.1752

Example 2: cosh(0) = (e⁰ + e⁻⁰) / 2 = (1 + 1) / 2 = 1

Example 3: tanh(2) = (e² - e⁻²) / (e² + e⁻²) ≈ 0.9640