Hyperbolic functions law

Definition 1 Hyperbolic Functions
sinh x = ex?e?x
2 cosh x = ex+e?x
2
tanh x = sinh x
cosh x coth x = cosh x
sinh x
sech x = 1
cosh x csch x = 1
sinh x
The hyperbolic functions are closely related to the trigonometric functions. The first example of this follows
from the identity
cosh2 x ? sinh2 x = 1
We can verify this from the definitions,
cosh2 x ? sinh2 x = e2x + e?2x + 2
4 ?
e2x + e?2x ? 2
4
= 1
Just like x = cos t and y = sin t gives a parametric representation for a circle, the above formula implies that
x = cosh t and y = sinh t gives a parametric representation for the right branch of the hyperbola x2?y2 = 1.
Just like their trigonometric counterparts, sinh x, tanh x, csch x, and coth x are odd functions, cosh x and
sech x are even functions. This is also easily verified from the definitions.
Definition 2 Derivatives of Hyperbolic Functions
Dx sinh x = cosh x Dx cosh x = sinh x
Dx tanh x = sech2 x Dx coth x = ?csch2 x
Dx sech x = ?sech x tanh x Dx csch x = ?csch x coth x
Another way the trigonometric and hyperbolic functions are connected concerns differential equations. The
functions sin x and cos x are solutions to the differential equation y00 + y = 0, while sinh x and cosh x are
solutions to the differential equation y00 ? y = 0.
Example 1 Find Dx cosh (tan x).
Solution
Dx cosh (tan x) = sinh (tan x)Dx (tan x)
= sec2 x · sinh (tan x)
1
Example 2 Find
R
coth x dx.
Solution Let u = sinh x, so that du = cosh x dx.
Z
coth x dx =
R cosh x
sinh x dx =
Z
1
u
du
= ln |u| + C = ln |sinh x| + C
Since hyperbolic sine and hyperbolic tangent have positive derivatives, they are increasing functions and
automatically have inverses. To obtain inverses for hyperbolic cosine and hyperbolic secant, we restrict their
domains to x  0. Thus,
x = sinh?1 y , y = sinh x
x = cosh?1 y , y = cosh x and x  0
x = tanh?1 y , y = tanh x
x = sech?1 y , y = sech x and x  0
Since the hyperbolic functions are defined in terms of exponentials, it seems likely that the inverse hyperbolic
functions can be defined in terms of the natural log, and indeed this is so as long as we make some necessary
domain restrictions.
sinh?1 x = ln
??
x +
p
x2 + 1


cosh?1 x = ln
??
x +
p
x2 ? 1


, x  1
tanh?1 x = 1
2 ln 1+x
1?x , ?