Mathematics reference
Hyperbolic trigonometry identities
|
|
|
Various identities essential in hyperbolic trigonometry.
|
|
|
Introduction.
|
- x and y are independent variables,
- e is the base of the natural logarithm.
- d is the differential operator,
- int is the integral symbol,
- C is the constant of integration.
|
|
Identities.
|
|
|
|
sinh x = (ex - e-x)/2
|
equation 1
|
|
|
|
cosh x = (ex + e-x)/2
|
equation 2
|
|
|
|
sech x = 1/cosh x
|
equation 3
|
|
|
|
csch x = 1/sinh x
|
equation 4
|
|
|
|
tanh x = sinh x/cosh x
|
equation 5
|
|
|
|
coth x = 1/tanh x
|
equation 6
|
|
|
|
cosh2 x - sinh2 x = 1
|
equation 7
|
|
|
|
tanh2 x + sech2 x = 1
|
equation 8
|
|
|
|
coth2 x - csch2 x = 1
|
equation 9
|
|
|
|
sinh (x + y) = sinh x cosh y + cosh x sinh y
|
equation 10
|
|
|
|
sinh (x - y) = sinh x cosh y - cosh x sinh y
|
equation 11
|
|
|
|
cosh (x + y) = cosh x cosh y + sinh x sinh y
|
equation 12
|
|
|
|
cosh (x - y) = cosh x cosh y - sinh x sinh y
|
equation 13
|
|
|
|
sinh2 x = [-1 + cosh (2 x)]/2
|
equation 14
|
|
|
|
sinh (2 x) = 2 sinh x cosh x
|
equation 15
|
|
|
|
cosh2 x = [1 + cosh (2 x)]/2
|
equation 16
|
|
|
|
cosh (2 x) = cosh2 x + sinh2 x
|
equation 17
|
|
|
|
arcsinh x = ln [x + (x2 + 1)1/2]
|
equation 18
|
|
|
|
arccosh x = ln [x + (x2 - 1)1/2]
|
equation 19
|
|
|
|
arctanh x = (1/2) ln [(1 + x)/(1 - x)]
|
equation 20
|
|
|
|
arccoth x = (1/2) ln [(x + 1)/(x - 1)]
|
equation 21
|
|
|
|
arcsech x = ln [[1 + (1 - x2)1/2]/x]
|
equation 22
|
|
|
|
arccsch x = ln [[1 + (1 + x2)1/2]/|x|]
|
equation 23
|
|
|
|
tanh (2 x) = 2 tanh x/(1 + tanh2 x)
|
equation 24
|
|
|
|
coth x - tanh x = 2 csch (2 x)
|
equation 25
|
|
|
|
coth x + tanh x = 2 coth (2 x)
|
equation 26
|
|
|
|
(d/dx) sinh x = cosh x
|
equation 27
|
|
|
|
(d/dx) cosh x = sinh x
|
equation 28
|
|
|
|
(d/dx) tanh x = sech2 x
|
equation 29
|
|
|
|
(d/dx) coth x = -csch2 x
|
equation 30
|
|
|
|
(d/dx) sech x = -sech x tanh x
|
equation 31
|
|
|
|
(d/dx) csch x = -csch x coth x
|
equation 32
|
|
|
|
(d/dx) arcsinh x = 1/(x2 + 1)1/2
|
equation 33
|
|
|
|
(d/dx) arccosh x = 1/(x2 - 1)1/2
|
equation 34
|
|
|
|
(d/dx) arctanh x = 1/(1 - x2)
|
equation 35
|
|
|
|
(d/dx) arccoth x = 1/(1 - x2)
|
equation 36
|
|
|
|
(d/dx) arcsech x = -1/[x (1 - x2)1/2]
|
equation 37
|
|
|
|
(d/dx) arccsch x = -1/[|x| (1 + x2)1/2]
|
equation 38
|
|
|
|
Integrals.
|
|
|
|
int cosh x dx = sinh x + C
|
equation 39
|
|
|
|
int sinh x dx = cosh x + C
|
equation 40
|
|
|
|
int sech2 x dx = tanh x + C
|
equation 41
|
|
|
|
int csch2 x dx = -coth x + C
|
equation 42
|
|
|
|
int sech x tanh x dx = -sech x + C
|
equation 43
|
|
|
|
int csch x coth x dx = -csch x + C
|
equation 44
|
|
|
|
int dx/(x2 + 1)1/2 = arcsinh x + C
|
equation 45
|
|
|
|
int dx/(x2 - 1)1/2 = arccosh x + C
|
equation 46
|
|
|
|
int dx/(1 - x2) = arctanh x + C
|
equation 47
|
|
|
|
int dx/(1 - x2) = arccoth x + C
|
equation 48
|
|
|
|
int dx/[x (1 - x2)1/2] = -arcsech x + C
|
equation 49
|
|
|
|
int dx/[|x| (1 + x2)1/2] = -arccsch x + C
|
equation 50
|
|
|
|
Navigation.
|
|
|
Erik Max Francis
-- TOP
Welcome to my homepage.
|
|
Reference
-- UP
A technical reference.
|
|
Mathematics reference
-- UP
A mathematics reference for students and teachers.
|
|
Mathematics reference: Trigonometric identities
-- PREVIOUS
Various identities and properties essential in trigonometry.
|
|
Mathematics reference: Limits
-- NEXT
Properties of limits.
|
|
|
Quick links.
|
|
|
Contents of Erik Max Francis' homepages
-- CONTENTS
Everything in my homepages.
|
|
Feedback
-- FEEDBACK
How to send feedback on these pages to the author.
|
|
About Erik Max Francis
-- PERSONAL
Information about me.
|
|
Copyright
-- COPYRIGHT
Copyright information regarding these pages.
|
|
|
|
Copyright © 2025 Erik Max Francis. All rights reserved.
|
|
|
|