The functions cosh x , sinh x and tanh x have much the same relationship to the rectangular hyperbola y2 = x2 - 1 as the circular functions do to the circle y2 = 1 - x2 . They are therefore sometimes called the hyperbolic functions (h for hyperbolic ).
Notation and pronunciation
cosh
{\displaystyle \displaystyle \cosh }
is an abbreviation for 'cosine hyperbolic', and
sinh
{\displaystyle \displaystyle \sinh }
is an abbreviation for 'sine hyperbolic'.
sinh
{\displaystyle \displaystyle \sinh }
is pronounced sinch ,
cosh
{\displaystyle \displaystyle \cosh }
is pronounced 'cosh', as you'd expect,
and
tanh
{\displaystyle \displaystyle \tanh }
is pronounced tanch .
[Diagram of rectangular hyperbola to illustrate]
Definitions [ edit ]
They are defined as
cosh
(
x
)
=
1
2
(
e
x
+
e
−
x
)
;
sinh
(
x
)
=
1
2
(
e
x
−
e
−
x
)
;
tanh
(
x
)
=
sinh
(
x
)
cosh
(
x
)
{\displaystyle \cosh(x)={\frac {1}{2}}(e^{x}+e^{-x});\,\,\sinh(x)={\frac {1}{2}}(e^{x}-e^{-x});\,\,\tanh(x)={\frac {\sinh(x)}{\cosh(x)}}}
Equivalently,
e
x
=
cosh
(
x
)
+
sinh
(
x
)
;
e
−
x
=
cosh
(
x
)
−
sinh
(
x
)
{\displaystyle \displaystyle e^{x}=\cosh(x)+\sinh(x);\,\,e^{-x}=\cosh(x)-\sinh(x)}
Reciprocal functions may be defined in the obvious way:
sech
(
x
)
=
1
cosh
(
x
)
;
cosech
(
x
)
=
1
sinh
(
x
)
;
coth
(
x
)
=
1
tanh
(
x
)
{\displaystyle \operatorname {sech} (x)={\frac {1}{\cosh(x)}};\,\,\operatorname {cosech} (x)={\frac {1}{\sinh(x)}};\,\,\coth(x)={\frac {1}{\tanh(x)}}}
1 - tanh2 (x) = sech2 (x); coth2 (x) - 1 = cosech2 (x)
It is easily shown that
cosh
2
(
x
)
−
sinh
2
(
x
)
=
1
{\displaystyle \displaystyle \cosh ^{2}(x)-\sinh ^{2}(x)=1}
, analogous to the result
cos
2
(
x
)
+
sin
2
(
x
)
=
1.
{\displaystyle \displaystyle \cos ^{2}(x)+\sin ^{2}(x)=1.}
In consequence, sinh(x) is always less in absolute value than cosh(x).
sinh(-x) = -sinh(x); cosh(-x) = cosh(x); tanh(-x) = -tanh(x).
Their ranges of values differ greatly from the corresponding circular functions:
cosh(x) has its minimum value of 1 for x = 0, and tends to infinity as x tends to plus or minus infinity;
sinh(x) is zero for x = 0, and tends to infinity as x tends to infinity and to minus infinity as x tends to minus infinity;
tanh(x) is zero for x = 0, and tends to 1 as x tends to infinity and to -1 as x tends to minus infinity.
[Add graph]
Addition formulae [ edit ]
There are results very similar to those for circular functions; they are easily proved directly from the definitions of cosh and sinh:
sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)
cosh(x+y) = cosh(x)cosh(y) + sinh(x)sinh(y)
Inverse functions [ edit ]
If y = sinh(x), we can define the inverse function x = sinh-1 y, and similarly for cosh and tanh. The inverses of sinh and tanh are uniquely defined for all x. For cosh, the inverse does not exist for values of y less than 1. For y = 1, x = 0. For y > 1, there will be two corresponding values of x, of equal absolute value but opposite sign. Normally, the positive value would be used. From the definitions of the functions,
sinh
−
1
x
=
ln
(
x
+
x
2
+
1
)
{\displaystyle \displaystyle \sinh ^{-1}x=\ln(x+{\sqrt {x^{2}+1}})}
cosh
−
1
x
=
ln
(
x
+
x
2
−
1
)
{\displaystyle \displaystyle \cosh ^{-1}x=\ln(x+{\sqrt {x^{2}-1}})}
tanh
−
1
x
=
1
2
ln
(
1
+
x
1
−
x
)
{\displaystyle \tanh ^{-1}x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}
Simplifying a cosh(x) + b sinh(x) [ edit ]
If a > |b| then
a
cosh
(
x
)
+
b
sinh
(
x
)
=
a
2
−
b
2
cosh
(
x
+
c
)
{\displaystyle \displaystyle a\cosh(x)+b\sinh(x)={\sqrt {a^{2}-b^{2}}}\cosh(x+c)}
where :
tanh
(
c
)
=
b
a
{\displaystyle \displaystyle \tanh(c)={\frac {b}{a}}}
If |a| < b then
a
cosh
(
x
)
+
b
sinh
(
x
)
=
b
2
−
a
2
sinh
(
x
+
d
)
{\displaystyle \displaystyle a\cosh(x)+b\sinh(x)={\sqrt {b^{2}-a^{2}}}\sinh(x+d)}
where :
tanh
(
d
)
=
a
b
{\displaystyle \displaystyle \tanh(d)={\frac {a}{b}}}
Relations to complex numbers [ edit ]
cos
(
i
x
)
=
cosh
(
x
)
{\displaystyle \cos(ix)=\cosh(x)}
cos
(
x
)
=
cosh
(
i
x
)
{\displaystyle \cos(x)=\cosh(ix)}
sin
(
i
x
)
=
i
sinh
(
x
)
{\displaystyle \sin(ix)=i\sinh(x)}
i
sin
(
x
)
=
sinh
(
i
x
)
{\displaystyle i\sin(x)=\sinh(ix)}
tan
(
i
x
)
=
i
tanh
(
x
)
{\displaystyle \tan(ix)=i\tanh(x)}
i
tan
(
x
)
=
tanh
(
i
x
)
{\displaystyle i\tan(x)=\tanh(ix)}
The addition formulae and other results can be proved from these relationships.
The gudermannian [ edit ]
The gudermannian (named after Christoph Gudermann, 1798–1852) is defined as gd(x) = tan-1 (sinh(x)). We have the following properties:
gd(0) = 0;
gd(-x) = -gd(x);
gd(x) tends to 1 ⁄2 π as x tends to infinity, and -1 ⁄2 π as x tends to minus infinity.
The inverse function gd-1 (x) = sinh-1 (tan(x)) = ln(sec(x)+tan(x)).
Differentiation [ edit ]
As can be proved from the definitions above,
d
d
x
sinh
(
x
)
=
cosh
(
x
)
;
d
d
x
cosh
(
x
)
=
sinh
(
x
)
;
d
d
x
tanh
(
x
)
=
s
e
c
h
2
(
x
)
;
d
d
x
g
d
(
x
)
=
s
e
c
h
(
x
)
{\displaystyle {\frac {d}{dx}}\sinh(x)=\cosh(x);\,\,{\frac {d}{dx}}\cosh(x)=\sinh(x);\,\,{\frac {d}{dx}}\tanh(x)=sech^{2}(x);\,\,{\frac {d}{dx}}gd(x)=sech(x)}
We also have
d
d
x
sinh
−
1
(
x
)
=
1
x
2
+
1
;
d
d
x
cosh
−
1
(
x
)
=
1
x
2
−
1
;
d
d
x
tanh
−
1
(
x
)
=
1
1
−
x
2
;
d
d
x
g
d
−
1
(
x
)
=
sec
(
x
)
{\displaystyle {\frac {d}{dx}}\sinh ^{-1}(x)={\frac {1}{\sqrt {x^{2}+1}}};\,\,{\frac {d}{dx}}\cosh ^{-1}(x)={\frac {1}{\sqrt {x^{2}-1}}};\,\,{\frac {d}{dx}}\tanh ^{-1}(x)={\frac {1}{1-x^{2}}};\,\,{\frac {d}{dx}}gd^{-1}(x)=\sec(x)}
.