Applications of Hyperbolic Functions Greg Kelly, Hanford High School, Richland, Washington

Applications of
Hyperbolic Functions
Greg Kelly, Hanford High School, Richland, Washington
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
Length of curve calculation:

d
c

d
c
2
 dy 
1    dx
 dx 
2 x
1  sinh  dx
a

d
c

d
c
 x
cosh  dx
a
2
x
cosh   dx
a
d
 x
a sinh  
 a c

Another example of a catenary is the Gateway Arch in St.
Louis, Missouri.

Another example of a catenary is the Gateway Arch in St.
Louis, Missouri.

If air resistance is
proportional to the square
of velocity:
y  A ln  cosh Bt 
y is the distance the
object falls in t seconds.
A and B 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

A third application is the tractrix.
(pursuit curve)
a
Both of these situations (and
others) can be modeled by:
 x
y  a sech    a 2  x 2
a
1
semi-truck
a
boat

The word tractrix comes from the Latin tractus, which
means “to draw, pull or tow”. (Our familiar word
“tractor” comes from the same root.)
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.
p