5.9 Hyperbolic Functions

Graph the following two functions:

2 2

x x x xe e e ey y

These functions show up frequently enough that theyhave been given names.

The behavior of these functions shows such remarkableparallels to trig functions, that they have been given similar names.

Hyperbolic Sine: sinh2

x xe ex

(pronounced “cinch x”)

Hyperbolic Cosine:

(pronounced “kosh x”)

cosh2

x xe ex

sinh coshx x 2

2

xe xe

2 2

x x x xe e e e

Note:

Definitions

2 2cosh sinh 1x x 2 2

2 2

x x x xe e e e

2 2 2 22 2

4 4

x x x xe e e e

41

4

Show that

Examples

2 2cosh sinh 1

Note that this is similar to but not the same as:

2 2sin cos 1

Now, if we have “trig-like” functions, it follows that we will have “trig-like” identities.

Identities

Hyperbolic Tangent:

sinhtanh

cosh

x x

x x

x e ex

x e e

“tansh (x)”

Hyperbolic Cotangent:

coshcoth

sinh

x x

x x

x e ex

x e e

“cotansh (x)”

Hyperbolic Secant: 1 2

sechcosh x x

xx e e

“sech (x)”

Hyperbolic Cosecant: 1 2

cschsinh x x

xx e e

“cosech (x)”

Definitions

2 2cosh sinh 1x x

Derive some hyperbolic trig identities from the following basic identity.

sin 2 2sin cosx x xDerive the double-angle identity, analogous to

Identities

sinh cosh2 2

x x x xd d e e e ex x

dx dx

cosh sinh2 2

x x x xd d e e e ex x

dx dx

Surprise, this is positive!

Derivaties

tanhx x

x x

d d e ex

dx dx e e

2

x x x x x x x x

x x

e e e e e e e e

e e

2 2 2 2

2

2 2x x x x

x x

e e e e

e e

2

4x xe e

22

x xe e

2sech x

(quotient rule)

2coth cschd

x xdx

sech sech tanhd

x x xdx

csch csch cothd

x x xdx

All derivatives are similar to trig functions except for some of the signs:

Sinh, Cosh and Tanh are positive. The others are negative.

Even though it looks like a parabola, it is not a parabola!

A hanging cable makes a shape called a catenary.

coshx

y b aa

(for some constant a)

sinhdy x

dx a

Another example of a catenary is the Gateway Arch in St. Louis, Missouri.

boat

semi-truck

A third application is the tractrix.

An example of a real-life situation that can be modeled by a tractrix equation is a semi-truck turning a corner.Another example is a boat attached to a rope being pulled by a person walking along the shore.

1 2 2 sechx

y a a xa

Other examples of a tractrix curve include a dog leaving the front porch and chasing person running on the sidewalk.

(pursuit curve)

Both of these situations (and others) can be modeled by: