Definition of hyperbolic functions

Hyperbolic sine of x$ extsinh x = frace^x - e^-x2$

Hyperbolic cosine of x$ extcosh x = frace^x + e^-x2$

Hyperbolic tangent of x$ exttanh x = frace^x - e^-xe^x + e^-x$

Hyperbolic cotangent of x$ extcoth x = frace^x + e^-xe^x - e^-x$

Hyperbolic secant of x$ extsech x = frac2e^x + e^-x$

Hyperbolic cosecant of x$ extcsch x = frac2e^x - e^-x$




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Relationship among hyperbolic functions

$ exttanh x = frac extsinh x extcosh x$

$ extcoth x = frac1 exttanh x = frac extcosh x extsinh x$

$ extsech x = frac1 extcosh x$

$ extcsch x = frac1 extsinh x$

$ extcosh^2x - extsinh^2x = 1$

$ extsech^2x + exttanh^2x = 1$

$ extcoth^2x - extcsch^2x = 1$

Functions of negative arguments

sinh(-x) = -sinh x

cosh(-x) = cosh x

tanh(-x) = -tanh x

csch(-x) = -csch x

sech(-x) = sech x

coth(-x) = -coth x

Addition fomulas

$ extsinh(x pm y) = extsinh x extcosh y pm extcosh x extsinh y$

$ extcosh(x pm y) = extcosh x extcosh y pm extsinh x extsinh y$

$ exttanh(x pm y) = frac exttanh x pm exttanh y1 pm exttanh x cdot exttanh y$

$ extcoth(x pm y) = frac extcoth x extcoth y pm l extcoth y pm extcoth x$

Double angle formulas

$ extsinh 2x = 2 extsinh x extcosh x$

$ extcosh 2x = extcosh^2x + extsinh^2x= 2 extcosh^2x - 1 = 1 + 2 extsinh^2x$

$ exttanh 2x = frac2 exttanh x1 + exttanh^2x$

Half angle formulas

$sinh fracx2 = pm sqrtfraccosh x - 12$ <+ if x > 0, - if x 0, - if x Multiple angle formulas

$sinh 3x = 3 sinh x + 4 sinh^3 x$

$cosh 3x = 4 cosh^3 x - 3 cosh x$

$ anh 3x = frac3 anh x + anh^3 x1 + 3 anh^2x$

$sinh 4x = 8 sinh^3 x cosh x + 4 sinh x cosh x$

$cosh 4x = 8 cosh^4 x - 8 cosh^2 x + 1$

$ anh 4x = frac4 anh x + 4 anh^3 x1 + 6 anh^2 x + anh^4 x$

Powers of hyperbolic functions

$ extsinh^2 x = frac12 extcosh 2x - frac12$

$ extcosh^2 x = frac12 extcosh 2x + frac12$

$ extsinh^3 x = frac14 extsinh 3x - frac34 extsinh x$

$ extcosh^3 x = frac14 extcosh 3x + frac34 extcosh x$

$ extsinh^4 x = frac38 - frac12 extcosh 2x + frac18 extcosh 4x$

$ extcosh^4 x = frac38 + frac12 extcosh 2x + frac18 extcosh 4x$

Sum, difference & product

$ extsinh x + extsinh y = 2 extsinh frac12(x + y) extcosh frac12(x - y)$

$ extsinh x - extsinh y = 2 extcosh frac12(x + y) extsinh frac12(x - y)$

$ extcosh x + extcosh y = 2 extcosh frac12(x + y) extcosh frac12(x - y)$

$ extcosh x - extcosh y = 2 extsinh frac12(x + y) extsinh frac12(x - y)$

$ extsinh x extsinh y = frac12( extcosh(x + y) - extcosh (x - y))$

$ extcosh x extcosh y = frac12( extcosh(x + y) + extcosh (x - y))$

$ extsinh x extcosh y = frac12( extsinh(x + y) + extsinh (x - y))$


Expression of hyperbolic functions in terms of others

In the following we assume x > 0. If x Graphs of hyperbolic functions
y = sinh x

*

y = cosh x
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y = tanh x
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y = coth x
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y = sech x
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y = csch x
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