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

\$ 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

y = tanh x

y = coth x

y = sech x

y = csch x

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