Hyperbolic Functions Calculator

Compute all six hyperbolic functions (sinh, cosh, tanh, csch, sech, coth) with exponential definitions and step-by-step solutions.

Enter Input Value

sinh(x)
1.175201
(e^x - e^-x)/2
cosh(x)
1.543081
(e^x + e^-x)/2
tanh(x)
0.761594
sinh(x)/cosh(x)
csch(x)
0.850918
1/sinh(x)
sech(x)
0.648054
1/cosh(x)
coth(x)
1.313035
cosh(x)/sinh(x)

Detailed Results

Input Value
x = 1
e^1 = 2.718282, e^-1 = 0.367879
e^x 2.718282
e^(-x) 0.367879
sinh(x) = (e^x - e^-x) / 2 1.175201
cosh(x) = (e^x + e^-x) / 2 1.543081
tanh(x) = sinh(x) / cosh(x) 0.761594
csch(x) = 1 / sinh(x) 0.850918
sech(x) = 1 / cosh(x) 0.648054
coth(x) = cosh(x) / sinh(x) 1.313035

Step-by-Step Computation

sinh(1) = (e^1 - e^-1) / 2 = 1.175201

What Are Hyperbolic Functions?

Hyperbolic functions are analogs of the ordinary trigonometric (circular) functions, but defined using the hyperbola rather than the circle. Just as trigonometric functions parameterize a unit circle (x2 + y2 = 1), the hyperbolic functions parameterize a unit hyperbola (x2 - y2 = 1).

They arise naturally in many areas of mathematics, including calculus, differential equations, complex analysis, and physics. They are defined in terms of the exponential function e^x.

The Six Hyperbolic Functions

sinh(x) - Hyperbolic Sine

The odd part of the exponential function. Domain: all reals. Range: all reals.

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

cosh(x) - Hyperbolic Cosine

The even part of the exponential function. Domain: all reals. Range: [1, infinity).

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

tanh(x) - Hyperbolic Tangent

Ratio of sinh to cosh. Domain: all reals. Range: (-1, 1).

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

csch(x) - Hyperbolic Cosecant

Reciprocal of sinh. Domain: all reals except 0. Range: all reals except 0.

csch(x) = 1 / sinh(x)

sech(x) - Hyperbolic Secant

Reciprocal of cosh. Domain: all reals. Range: (0, 1].

sech(x) = 1 / cosh(x)

coth(x) - Hyperbolic Cotangent

Reciprocal of tanh. Domain: all reals except 0. Range: (-inf, -1) U (1, inf).

coth(x) = cosh(x) / sinh(x)

Key Identities

Fundamental Identity

The hyperbolic analog of the Pythagorean identity sin2(x) + cos2(x) = 1:

cosh2(x) - sinh2(x) = 1

Note the minus sign, which contrasts with the plus sign in the circular case.

Other Important Identities

  • sinh(-x) = -sinh(x) (odd function)
  • cosh(-x) = cosh(x) (even function)
  • sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y)
  • cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y)
  • e^x = cosh(x) + sinh(x)
  • e^(-x) = cosh(x) - sinh(x)
  • 1 - tanh2(x) = sech2(x)
  • coth2(x) - 1 = csch2(x)

Derivatives

  • d/dx sinh(x) = cosh(x)
  • d/dx cosh(x) = sinh(x)
  • d/dx tanh(x) = sech2(x)
  • d/dx csch(x) = -csch(x) coth(x)
  • d/dx sech(x) = -sech(x) tanh(x)
  • d/dx coth(x) = -csch2(x)

Applications

  • Catenary curves: A hanging chain or cable forms a catenary described by y = a cosh(x/a)
  • Special relativity: The rapidity parameter uses hyperbolic functions for Lorentz boosts
  • Engineering: Transmission line analysis, heat transfer, and signal processing
  • Machine learning: tanh is used as an activation function in neural networks
  • Complex analysis: Connecting trigonometric and hyperbolic functions via Euler's formula

Connection to Trigonometric Functions

Through complex numbers, the hyperbolic and trigonometric functions are intimately related:

  • sinh(ix) = i sin(x)
  • cosh(ix) = cos(x)
  • sin(ix) = i sinh(x)
  • cos(ix) = cosh(x)