Hyperbolic Function Facts—Pythagorean Theorem
Here’s another flurry of facts, this time involving the hyperbolic functions. You are not expected to memorize any of these, but you should be able to use them in calculations if you are given numbers to “plug in.”
Pythagorean Theorem For Sinh And Cosh
The difference between the squares of the hyperbolic sine and hyperbolic cosine of a variable is always equal to either 1 or –1. The following formulas hold for all real numbers x :
sinh ^{2} x = cosh ^{2} x = –1
cosh ^{2} x – sinh ^{2} x = 1
Pythagorean Theorem For Csch And Coth
The difference between the squares of the hyperbolic cotangent and hyperbolic cosecant of a variable is always equal to either 1 or –1. The following formulas hold for all real numbers x except 0:
csch ^{2} x – coth ^{2} x = –1
coth ^{2} x – csch ^{2} x = 1
Pythagorean Theorem For Sech And Tanh
The sum of the squares of the hyperbolic secant and hyperbolic tangent of a variable is always equal to 1. The following formula holds for all real numbers x :
sech ^{2} x + tanh ^{2} x = 1
Negative Variables
Hyperbolic Sine Of Negative Variable
The hyperbolic sine of the negative of a variable is equal to the negative of the hyperbolic sine of the variable. The following formula holds for all real numbers x:
sinh – x = –sinh x
Hyperbolic Cosine Of Negative Variable
The hyperbolic cosine of the negative of a variable is equal to the hyperbolic cosine of the variable. The following formula holds for all real numbers x :
cosh – x = cosh x
Hyperbolic Tangent Of Negative Variable
The hyperbolic tangent of the negative of a variable is equal to the negative of the hyperbolic tangent of the variable. The following formula holds for all real numbers x :
tanh – x = –tanh x
Hyperbolic Cosecant Of Negative Variable
The hyperbolic cosecant of the negative of a variable is equal to the negative of the hyperbolic cosecant of the variable. The following formula holds for all real numbers x except 0:
csch – x = –csch x
Hyperbolic Secant Of Negative Variable
The hyperbolic secant of the negative of a variable is equal to the hyperbolic secant of the variable. The following formula holds for all real numbers x :
sech – x = sech x
Hyperbolic Cotangent Of Negative Variable
The hyperbolic cotangent of the negative of a variable is equal to the negative of the hyperbolic cotangent of the variable. The following formula holds for all real numbers x except 0:
coth – x = –coth x
Double Values
Hyperbolic Sine Of Double Value
The hyperbolic sine of twice any given variable is equal to twice the hyperbolic sine of the original variable times the hyperbolic cosine of the original variable. The following formula holds for all real numbers x :
sinh 2 x = 2 sinh x cosh x
Hyperbolic Cosine Of Double Value
The hyperbolic cosine of twice any given variable can be found according to any of the following three formulas for all real numbers x :
cosh 2 x = cosh ^{2} x + sinh ^{2} x
cosh 2 x = 1 + 2 sinh ^{2} x
cosh 2 x = 2 cosh ^{2} x – 1

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