Math2.org Math Tables: Hyperbolic Trigonometric Identities

Hyperbolic Definitions

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

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 )

cosh²(x) - sinh²(x) = 1

tanh²(x) + sech²(x) = 1

coth²(x) - csch²(x) = 1

Inverse Hyperbolic Defintions

arcsinh(z) = ln( z + (z² + 1) )

arccosh(z) = ln( z (z² - 1) )

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

arccsch(z) = ln( (1+(1+z²) )/z )

arcsech(z) = ln( (1(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)