Hyperbolic Functions - Math Is Fun

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Hyperbolic Functions

The two basic hyperbolic functions are "sinh" and "cosh":

Hyperbolic Sine:

sinh(x) = ex − e-x 2

(pronounced "shine")

Hyperbolic Cosine:

cosh(x) = ex + e-x 2

(pronounced "cosh")

They use the natural exponential function ex

And are not the same as sin(x) and cos(x), but a little bit similar:

sinh vs sin function sinh vs sin

cosh vs cos function cosh vs cos

rope bridge

Catenary

One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains.

A hanging cable forms a curve called a catenary defined using the cosh function:

f(x) = a cosh(x/a)

Like in this example from the page arc length :

catenary graph

Other Hyperbolic Functions

From sinh and cosh we can create:

Hyperbolic tangent "tanh" (pronounced "than"):

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

tanh vs tan function tanh vs tan

Hyperbolic cotangent:

coth(x) = cosh(x) sinh(x) = ex + e-x ex − e-x

Hyperbolic secant:

sech(x) = 1 cosh(x) = 2 ex + e-x

Hyperbolic cosecant "csch" or "cosech":

csch(x) = 1 sinh(x) = 2 ex − e-x

Why the Word "Hyperbolic" ?

Because it comes from measurements made on a Hyperbola:

tanh vs tan function

So, just like the trigonometric functions relate to a circle, the hyperbolic functions relate to a hyperbola.

Identities

  • sinh(−x) = −sinh(x)
  • cosh(−x) = cosh(x)

And

  • tanh(−x) = −tanh(x)
  • coth(−x) = −coth(x)
  • sech(−x) = sech(x)
  • csch(−x) = −csch(x)

Odd and Even

Both cosh and sech are Even Functions, the rest are Odd Functions.

Derivatives

The Derivatives are:

d dx sinh(x) = cosh(x)

d dx cosh(x) = sinh(x)

d dx tanh(x) = 1 − tanh2(x)

Common Functions Reference Sets Index

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