Hyperbolic Trigonomic Identities

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Math2.org Math Tables: Hyperbolic Trigonometric Identities
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Hyperbolic Definitions

sinh(x) = ( ex - e-x )/2

csch(x) = 1/sinh(x) = 2/( ex - e-x )

cosh(x) = ( e x + e -x )/2

sech(x) = 1/cosh(x) = 2/( ex + e-x )

tanh(x) = sinh(x)/cosh(x) = ( ex - e-x )/( ex + e-x )

coth(x) = 1/tanh(x) = ( ex + e-x)/( ex - e-x )

cosh2(x) - sinh2(x) = 1

tanh2(x) + sech2(x) = 1

coth2(x) - csch2(x) = 1

Inverse Hyperbolic Defintions

arcsinh(z) = ln( z + [sqrt](z2 + 1) )

arccosh(z) = ln( z [sqrt](z2 - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+[sqrt](1+z2) )/z )

arcsech(z) = ln( (1[sqrt](1-z2) )/z )

arccoth(z) = 1/2 ln( (z+1)/(z-1) )

Relations to Trigonometric Functions

sinh(z) = -i sin(iz)

csch(z) = i csc(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)

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