5.9
Hyperbolic Functions
Graph the following two functions:
e e
y
2
x
x
e e
y
2
x
x
These functions show up frequently enough that they
have been given names.
The behavior of these functions shows such remarkable
parallels to trig functions, that they have been given
similar names.
Definitions
e e
sinh  x  
2
x
Hyperbolic Sine:
(pronounced “cinch x”)
e e
cosh  x  
2
x
Hyperbolic Cosine:
(pronounced “kosh x”)
Note:
x
x
x
e x  e x e x  e x
2e
x
sinh  x   cosh  x  

e

2
2
2
Examples
Show that
e e

2

x
x
cosh x  sinh x  1
2
2
 e e
 
2
 
e 2e

4
2x
2
2 x
x
x



2
e 2e

4
2x
2 x
4
 1
4
Identities
Note that this is similar to
but not the same as:
cosh 2   sinh 2   1
sin 2   cos 2   1
Now, if we have “trig-like”
functions, it follows that we
will have “trig-like” identities.
Definitions
Hyperbolic Tangent:
“tansh (x)”
sinh  x 
e x  e x
tanh  x  
 x x
cosh  x  e  e
cosh  x 
e x  e x
 x x
Hyperbolic Cotangent: coth  x  
sinh  x  e  e
“cotansh (x)”
Hyperbolic Secant:
“sech (x)”
Hyperbolic Cosecant:
“cosech (x)”
1
2
sech  x  
 x x
cosh  x  e  e
1
2
csch  x  
 x x
sinh  x  e  e
Identities
Derive some hyperbolic trig identities from the following
basic identity.
cosh 2 x  sinh 2 x  1
Derive the double-angle identity, analogous to
sin 2x  2sin x cos x
Derivaties
d
d e e
sinh  x  
dx
dx
2
x
x
e e

2
x
d
d e e
cosh  x  
dx
dx
2
x
e e

2
x
x
x
x
Surprise, this is positive!
 cosh  x 
 sinh  x 
x
d
d e e
tanh  x  
dx
dx e x  e x
x
(quotient
rule)
e




x
 e  x  e x  e  x    e x  e  x  e x  e  x 
2x
e

e  e 
 2  e   e  2  e 
e  e 
x
x
2 x
x
4
x
x
e

e


2
2
2 x
2x
x
2
2


 x
x 
e

e


2
 sech 2  x 
d
coth  x   csch 2  x 
dx
d
sech  x   sech  x  tanh  x 
dx
d
csch  x   csch  x  coth  x 
dx
All derivatives are similar to trig functions except for some
of the signs:
 Sinh, Cosh and Tanh are positive.
 The others are negative.
A hanging cable makes a shape called a catenary.
x
yEven
 b though
a coshit looks

 not
like a parabola, ita is
a parabola!
(for
some constant a)
dy
x
 sinh  
dx
a
Another example of a catenary
is the Gateway Arch in St.
Louis, Missouri.
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.
Both of these situations (and others)
can be modeled by:
semi-truck
 x
2
2
y  a sech    a  x
a
1
Other examples of a tractrix curve
include a dog leaving the front porch
and chasing person running on the
sidewalk.
boat