Hyperbolic Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2005

Scotty’s Castle, Death Valley, CA

Consider 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.

2 2

x x x xe e e ey y

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

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)”

First, an easy one:

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

sinh coshx x

sinh cosh xx x e

2

2

xe

xe

2 2

x x x xe e e e

(This one doesn’t really have an analogy in trig.)

2 2cosh sinh 1x x 2 2

12 2

x x x xe e e e

2 2 2 22 2

14 4

x x x xe e e e

41

4

1 1

2 2cosh sinh 1x x

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

2 2sin cos 1x x

There are several other identities in table A6.2 on page 619.

I will give you a sheet with the formulas on it to use on the test.

Derivatives can be found relatively easily using the definitions.

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!

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 of the derivatives are similar to trig functions except for some of the signs.Sinh, Cosh and Tanh are positive.The others are negative

Integral formulas can be written from the derivative formulas.(See the table on page 620.)

Or you can use the catalog.

2nd MATH C:Hyperbolic

On the TI-89, the hyperbolic functions are under: