Hyperbolic Functions

Consider the following two functions:

y



e x



e



x

2

y



e x



e



x

2 These functions show up frequently enough that they have been given names.



y



e x



e



x

2

y



e x



e



x

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



Hyperbolic Sine: sinh 

e x



e



x

2 (pronounced “cinch x”) Hyperbolic Cosine: (pronounced “kosh x”) cosh 

e x



e



x

2 

Hyperbolic Tangent: “tansh (x)” tanh Hyperbolic Cotangent: coth “cotansh (x)”  sinh cosh     

e e x x



e



x



e



x

 cosh sinh     

e e x x



e



x



e



x

Hyperbolic Secant: “sech (x)” sech  1 cosh 

e x

2 

e



x

Hyperbolic Cosecant: “cosech (x)” csch  sinh 1   

e x

2 

e



x



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

First, an easy one: 

sinh sinh  cosh  cosh 

e x



e



x

2 

e x



e



x

2  2

e x

2 

e x



e x

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

cosh 2

x

 sinh 2

x

 1

e

2

x

 

e x



e



x

2   2

e

 2

x

 4   

e

2

e x x



e



x

2   2

e

 2

x

 1  1 4 4  1 4 Note that this is similar to but not the same as: sin 2

x

 cos 2

x

 1 

Derivatives can be found relatively easily using the definitions.

d dx

sinh 

d e dx x



e



x

2 

e x



e



x

2  cosh

d dx

cosh 

d e dx x



e



x

2 

e x



e



x

2  sinh Surprise, this is positive!



d dx

tanh (quotient rule)  

d e x



e



x



dx e x e x

 

e



x e



x



e x

 

e

2

x e

 2

x



e x



e



x



e x



e



x e x

 2 

e



x



e x e

2

x



e



x

 2

e

 2

x

 

e



x

  

e x

4 

e



x

 2

e x

 2

e



x

2  sech 2 

d dx

coth

d dx

sech     sech csch 2 tanh

d dx

csch   csch coth 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 

Applications of Hyperbolic Functions

A hanging cable makes a shape called a catenary.

y a

  a parabola!

a

)

dy dx

 sinh

x

  Length of curve calculation: 

c d

1 

dx

  2

dx



c d

cosh 2

x

 

dx



c d

2

x

 

dx



c d

cosh

x

 

dx a

sinh

x

 

d c



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



Taking Air Resistance into consideration, free-fall can be calculated by:

y



A



Bt



y

is the distance the object falls in

t A

and

B

seconds.

are constants.



A third application is the tractrix.

(pursuit curve) 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.

semi-truck boat



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

y



a

sech  1

x

  

a

2 

x

2

a a

semi-truck boat



The word tractrix comes from the Latin “tractor” comes from the same root.)

tractus

, which means “to draw, pull or tow”. (Our familiar word Other examples of a tractrix curve include a heat-seeking missile homing in on a moving airplane, and a dog leaving the front porch and chasing person running on the sidewalk.

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