Hyperbolic Functions
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Hyperbolic Functions in Algebra and Calculus
By Kevin Mirus, Madison Area Technical College
1. Uses of the hyperbolic functions:
a. Engineering applications (catenary curves)
b. Finding certain antiderivatives.
c. Solving differential equations, such as
y(x) = y(x), which has solution y(x) = Asinh(x) + Bcosh(x)
vs. y(x) = -y(x), which has solution y(x) = Asin(x) + Bcos(x)
2. Definition of the hyperbolic functions:
e x  ex
2
sinh x  e x  e  x e 2 x  1 


tanh x 

cosh x  e x  e  x e 2 x  1 
1
2
sech x 
 x
cosh x e  e  x
sinh x 
e x  ex
2
cosh x  e x  e  x e 2 x  1 


coth x 

sinh x  e x  e  x e 2 x  1 
1
2
csch x 
 x
sinh x e  e  x
cosh x 
3. Some identities involving hyperbolic functions:
cosh 2 u  sinh 2 u  1
Note: if x = cosh u and y = sinh u, then
x 2  y 2  1 is the equation of a
hyperbola
sinh 2u  2 sinh u cosh u
cosh (s + t) = cosh s cosh t + sinh s sinh t
sinh  i  

ei  e i
2
 cos  i sin     cos     i sin    
2
 i sin 
cosh 2u  2 cosh 2 u  1
sinh (s + t) = sinh s cosh t + cosh s sinh t
4. Osbornes’s Rule: a trigonometry identity can be converted to an analogous identity for
hyperbolic functions by expanding, exchanging trigonometric functions with their
hyperbolic counterparts, and then flipping the sign of each term involving the product of
two hyperbolic sines. For example, given the identity
Osborne's rule gives the corresponding identity
Citation: Eric W. Weisstein. "Osborne's Rule." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/OsbornesRule.html
Hyperbolic Functions
5. Graphs of the hyperbolic functions:
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Hyperbolic Functions
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6. Inverses of the hyperbolic functions and their formulae:
y  sinh 1 x iff sinh y  x
y  cosh 1 x iff cosh y  x

sinh 1 x  ln x  x 2  1
y  tanh
1
tanh 1 x 


cosh 1 x  ln x  x 2  1
1
x iff tanh y  x
y  coth
1 1 x
ln
2 1 x
coth 1 x 

x iff coth y  x
1 x 1
ln
2 x 1
y  sech 1 x iff sech y  x
1
Also: coth 1  x   tanh 1  
 x
1
y  csch x iff csch y  x
1
sech 1 x  ln  
x
1
csch 1 x  ln  
x

1
 1 
2
x

1
Also: sech 1  x   cosh 1  
 x

1
 1 
2
x

1
Also: csch 1  x   sinh 1  
 x
7. Proof of the formula for the inverse hyperbolic sine:
y  sinh 1 x
x  sinh y 
e y  ey
2
2x  e y  e y
2 xe y  e 2 y  1
e 2 y  2 xe y  1  0
e 
y 2
 
 2x e y  1  0
2 x  4 x 2  4(1)( 1)
e 
 x  x2 1
2(1)
y

y  sinh 1 x  ln x  x 2  1

Hyperbolic Functions
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8. Composition of hyperbolic functions and inverse hyperbolic functions:
(Sorry this is so incomplete right now)
sinh  sinh 1  x    x
cosh  sinh 1  x   
tanh  sinh 1  x   
sinh  cosh 1  x    x 2  1
cosh  cosh 1  x    x
tanh  cosh 1  x   
x
cosh  tanh 1  x   
tanh  tanh 1  x    x
1  x2
cosh  coth 1  x   
tanh  coth 1  x   
cosh  sech 1  x   
tanh  sech 1  x   
cosh  csch 1  x   
tanh  csch 1  x   
sech  sinh 1  x   
csch  sinh 1  x   
sech  cosh 1  x   
csch  cosh 1  x   
sech  tanh 1  x   
csch  tanh 1  x   
sech  coth 1  x   
csch  coth 1  x   
sech  sech 1  x    x
csch  sech 1  x   
sech  csch 1  x   
csch  csch 1  x    x
sinh  tanh 1  x   
sinh  coth 1  x   
sinh  sech
 x  
sinh  csch 1  x   
coth  sinh 1  x   
coth  cosh 1  x   
coth  tanh 1  x   
coth  coth 1  x    x
coth  sech 1  x   
coth  csch 1  x   
1
9. Derivatives of the hyperbolic functions:
d
d e x  ex e x  ex
sinh x 

 cosh x
dx
dx
2
2
d
tanh x  sech 2 x
dx
d
sech x  sech x tanh x
dx
d
cosh x  sinh x
dx
d
coth x  - csch 2 x
dx
d
csch x  csch x coth x
dx
10. Recall that the derivative formulas can be used to obtain integral formulas, like
 sinh xdx  cosh x  c
Hyperbolic Functions
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11. Derivatives of the inverse hyperbolic functions:
d
1
d
sinh 1 x 
cosh 1 x 
2
dx
dx
1 x
d
1
tanh 1 x 
dx
1- x2
d
-1
sech -1 x 
dx
x 1- x 2
1
x2 1
d
1
coth 1 x 
dx
1- x2
d
-1
csch -1 x 
dx
x 1 x2
12. Recall that the derivative formulas can be used to obtain integral formulas, like
1
1
 1  x 2 dx  sinh x  c
Also, you can prove using integration by substitution that
 tanh  x  dx  ln cosh  x   c , and that
 coth  x  dx  ln sinh  x   c
13. Proof of the derivative of the inverse hyperbolic sine:
d
d
y  sinh 1 x
sinh 1 x  ln x  x 2  1
dx
dx
x  sinh y
1
d
d
d

  x  x 2  1
sinh y 
x
2

x  x  1 dx 
dx
dx
x
dy
1
cosh y
1
2
2
x 1  x  x 1
dx

dy
1
1
x  x2  1 x  x2  1


dx cosh y
x2
1  sinh 2 y
x  x2  1 
x
2
x

1
dy
1


x 2   x 2  1
dx
1 x2


 x2  1 

1
 x  1  x 2
2


x2  1
1
1  x2
x2
x2  1