Transcript Calculus 11.1 - University of Houston

Hyperbolic Functions
Scotty’s Castle, Death Valley, CA
Photo by Vickie Kelly, 2005
Greg Kelly, Hanford High School, Richland, Washington
Consider 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.

e e
y
2
x
x
e e
y
2
x
x
The behavior of these functions shows such remarkable
parallels to trig functions, that they have been given
similar names.

e e
sinh  x  
2
x
Hyperbolic Sine:
x
(pronounced “cinch x”)
e e
cosh  x  
2
x
Hyperbolic Cosine:
x
(pronounced “kosh x”)

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)”
1
2
sech  x  
 x x
cosh  x  e  e
1
2
 x x
Hyperbolic Cosecant: csch  x  
sinh  x  e  e
“cosech (x)”

Now, if we have “trig-like” functions, it follows that
we will have “trig-like” identities.
First, an easy one:

e e
sinh  x   cosh  x  
2
x
2e

2
x
e e

2
x
x
x
 ex
sinh  x   cosh  x   e
x
(This one doesn’t really have an analogy in trig.)

cosh 2 x  sinh 2 x  1
2
2
 e e   e e 

 
 1
 2   2 
2x
2 x
2x
2 x
e 2e
e 2e

1
4
4
4
1
4
x
x
x
x
11

cosh 2 x  sinh 2 x  1
Note that this is similar to but not the same as:
sin 2 x  cos2 x  1
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.
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
 cosh  x 
 sinh  x 
Surprise, this is positive!

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
 sech2  x 

d
2
coth  x   csch  x 
dx
d
sech  x   sech  x  tanh  x 
dx
d
csch  x   csch  x  coth  x 
dx
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.)
On the TI-89, the hyperbolic functions are under:
2nd
MATH
C:Hyperbolic
Or you can use the catalog.
p