Hyperbolic Trigonometric Identities

Hyperbolic Definitions

csch(x) = 1/sinh(x) = 2/( e^{x} - e^{-x} )

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

sech(x) = 1/cosh(x) = 2/( e^{x} + e^{-x} )

tanh(x) = sinh(x)/cosh(x) = ( e^{x} - e^{-x} )/( e^{x} + e^{-x} )

coth(x) = 1/tanh(x) = ( e^{x} + e^{-x})/( e^{x} - e^{-x} )

cosh^{2}(x) - sinh^{2}(x) = 1

tanh^{2}(x) + sech^{2}(x) = 1

coth^{2}(x) - csch^{2}(x) = 1

Inverse Hyperbolic Defintions

arcsinh(z) = ln( z + [sqrt](z^{2} + 1) )

arccosh(z) = ln( z [sqrt](z^{2} - 1) )

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

arccsch(z) = ln( (1+[sqrt](1+z^{2}) )/z )

arcsech(z) = ln( (1[sqrt](1-z^{2}) )/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)