Hyperbolic functions
Index
FAQ
Hyperbolic functions
Hungarian and English notation
e e
sinh( x) 
2
x
x
e e
cosh( x) 
2
x
x
x
x
sinh x e  e
e e
tanh( x ) 
 x
coth( x )  x
x
cosh x e  e
e  ex
x
Index
x
FAQ
Groupwork in 4 groups
For each function :
- find domain
- discuss parity
- find limits at the endpoints of the domain
-find zeros if any
-find intervals such that the function is cont.
-find local and global extremas if any
-find range
-find asymptotes
Index
FAQ
Summary: cosh
e e
cosh( x) 
2
x
x
What are the asymptotes of
cosh(x)
-in the infinity (2)
-negative infiniy (2)
PROVE YOUR STATEMENT!
Index
EVEN : cosh h ( x ) FAQ
 cosh x
Summary: cosh

Application of the use of hyperbolic cosine to
describe the shape of a hanging wire/chain.
Index
FAQ
Background
So, cables like power line cables, which hang freely,
hang in curves called hyperbolic cosine curves.
Index
FAQ
Chaincurve-catentity
Index
FAQ
Background
Suspension cables like
those of the Golden Gate
Bridge, which support a
constant load per horizontal
foot, hang in parabolas.
Index
FAQ
Which shape do you suppose in
this case?
Index
FAQ
Application: we will solve it
SOON!

Electric wires suspended between two
towers form a catenary with the equation
x
y  60 cosh
60

120'
If the towers are 120 ft apart, what is the
length of the suspended wire?
• Use the arc length formula
b
L   1   f '( xi )  dx
2
a
Index
FAQ
Summary: sinh
e e
sinh( x ) 
2
x
x
What are the asymptotes of
cosh(x)
-in the infinity (2)
-negative infiniy (2)
PROVE YOUR STATEMENT!
Index
ODD : sinh(  x )  FAQ
sinh x
Analogy between trigonometric
and hyperbolic functions
If t is any real number, then the point
P(cos t, sin t) lies on the unit circle
x2 + y2 = 1 because cos2 t + sin2 t = 1.
T is the OPQ angle measured in
radian Trigonometric functions are
also called CIRCULAR functions
Index
If t is any real number, then
the point P(cosh t, sinh t) lies on the
right branch of the hyperbola
x2 - y2 = 1 because cosh2 t - sin2 t = 1
and cosh t ≥ 1. t does not represent
the measure of an angle.
HYPERBOLIC functions
FAQ
HYPERBOLIC FUNCTIONS
In the trigonometric case t
represents twice the
area of the shaded
circular sector
Index
It turns out that t represents twice
the area of the shaded
hyperbolic sector
FAQ
Identities
sinh  x   cosh  x   e
x
Except for the one above. if we have “trig-like” functions, it follows
that we will have “trig-like” identities. For example:
sin x  cos x  1
cosh x  sinh x  1
sin 2 x  2 sin x cos x
sinh 2 x  2 sinh x cosh x
cos 2 x  cos 2 x  sin 2 x
cosh 2x  cosh 2 x  sinh 2 x
2
Index
2
2
2
FAQ
Proof of
cosh x  sinh x  1
2
2
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
Index
x
x
x
11
FAQ

Other identities
HW: Prove all remainder ones in your cheatsheet!
Index
FAQ

Derivatives
d
d e e
sinh  x  
dx
dx
2
x
x
e e

2
x
x
 cosh  x 
d
d e x  e x e x  e x
cosh  x  

 sinh  x 
dx
dx
2
2
Surprise, this is positive!
Index
FAQ

Summary: Tanh(x)
What are the asymptotes
of tanh(x)
-in the infinity (2)
-In the negative infiniy (2)
PROVE YOUR
STATEMENT!
Find the derivative!
Index
x
sinh( x ) e  e
tanh( x ) 
 x x
cosh( x ) e  e
x
FAQ
Application of tanh: description
of ocean waves
The velocity of a water wave with length L moving across
a body of water with depth d is modeled by
the function gL
 2 d 
v
tanh 

2
 L 
where g is the acceleration due to gravity.
Index
FAQ
Hyperbolic cotangent
cosh  x  e x  e  x
coth  x  
 x x
sinh  x  e  e
What are the asymptotes of
cotanh(x)
-in the infinity (2)
-In the negative infiniy (2)
-At 0?
PROVE YOUR STATEMENT!
Find the derivative!
Index
FAQ
Summary: Hyperbolic
Functions
Index
FAQ
INVERSE HYPERBOLIC FUNCTIONS
The sinh is one-to-one function. So, it has
inverse function denoted by sinh-1
Index
FAQ
INVERSE HYPERBOLIC FUNCTIONS
The tanh is one-to-one function. So, it has
inverse function denoted by tanh-1
Index
FAQ
INVERSE FUNCTIONS
This figure shows that cosh
is not one-toone.However, when
restricted to the domain
[0, ∞],
it becomes one-to-one.
The inverse hyperbolic
cosine function is defined
as the inverse
of this restricted function
Index
FAQ
Inverse hyperbolic functions
HW.: Define the inverse of the coth(x) function
Index
FAQ
INVERSE FUNCTIONS
1

x  ln  x 

 1
sinh x  ln x  x  1
cosh
1
2
x
2
 1 x 
tanh x  ln 

 1 x 
1
Index
1
2
x
x 1
1 x  1
FAQ
INVERSE FUNCTIONS

sinh 1 x  ln x  x 2  1

e y  e y
x  sinh y 
2
y 2
y
(e ) – 2x(e ) – 1 = 0
ey – 2x – e-y = 0
multiplying by ez . e2y – 2xey – 1 = 0
(ey)2 – 2x(ey) – 1 = 0
2x  4 x2  4
e 
 x  x2  1
2
y

sinh 1 x  ln x  x 2  1
Index

x  x2  1  0
ey >0
FAQ
DERIVATIVES
The formulas for the derivatives of
d
1
1
(sinh x) 
2
dx
1 x
tanh-1x and coth-1x appear to be
d
1
cosh
x 

dx
functions have no numbers in common:
1
x2  1
d
1
1
(tanh x) 
2
dx
1 x
Index
d
1
1
identical.
(csc h x)  
2
dx the domains of these
However,
x x 1
d
1
1
(sec h x) 
tanh-1x is defined for | x | < 1. 2
dx
x
coth-1x is defined for x| x |1>1.
d
1
1
(coth x) 
2
dx
1 x
FAQ
Sources:

http://www.mathcentre.ac.uk/resources/
workbooks/mathcentre/hyperbolicfunctio
ns.pdf
Index
FAQ