Hyperbolic Trig Functions

Greg Kelly, Hanford High School, Richland, Washington

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

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.

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.

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

2nd MATH Hyperbolic

Or you can use the catalog.

Teacher find the derivative of (ex + e-x)/2

Student: Oh that’s a cinch