Hyperbolic Functions

The two basic hyperbolic functions are:

sinh and cosh
(pronounced "shine" and "cosh")

sinh x = ex − e−x 2

cosh x = ex + e−x 2

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

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: 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

Derivatives are:

d dx sinh x = cosh x

d dx cosh x = sinh x

d dx tanh x = 1 − tanh2 x