Hyperbolic Functions - Math Is Fun

Hyperbolic Functions

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

Hyperbolic Sine:

sinh(x) = ex − e-x2

(pronounced "shine" or "sinch")

Hyperbolic Cosine:

cosh(x) = ex + e-x2

(pronounced "kosh")

They use the natural exponential function ex

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

sinh vs sin

cosh vs cos

Note: cosh(x) is always positive, and cosh(0)=1

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 :

Here a is a positive constant: larger a makes the curve wider and flatter, and smaller a makes it narrower and steeper.

Other Hyperbolic Functions

From sinh and cosh we can create:

Hyperbolic tangent "tanh":

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

tanh vs tan

(pronounced "tanch" or "than")

Hyperbolic cotangent:

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

not defined at x=0 because that would divide by 0

Hyperbolic secant:

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

Hyperbolic cosecant "csch" or "cosech":

csch(x) = 1sinh(x) = 2ex − e-x

not defined at x=0 because that would divide by 0

Why the Word "Hyperbolic" ?

Because it comes from measurements made on a Hyperbola:

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)

And the very important identity

• cosh2(x) − sinh2(x) = 1

(compare with cos2(x) + sin2(x) = 1)

Pronounciation Guide

People have a lot of fun pronouncing these functions!

Function	Common Pronunciation	Rhymes With...
sinh(x)	"shine" or "sinch"	mine / pinch
cosh(x)	"kosh"	gosh
tanh(x)	"tanch" or "than"	ranch / ran
sech(x)	"seech" or "setch"	beach / fetch
csch(x)	"ko-seech"	(no direct rhyme)
coth(x)	"koth"	both (with a soft 'th')

Odd and Even

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

Derivatives

The Derivatives are:

ddx sinh(x) = cosh(x)

ddx cosh(x) = sinh(x)

ddx tanh(x) = 1 − tanh2(x)

Trigonometric Identities Common Functions Reference Sets Index