Hyperbolic Function - Useful Relations

Useful Relations

Odd and even functions:

\begin{align} \sinh(-x) &= -\sinh (x) \\ \cosh(-x) &= \cosh (x) \end{align}

Hence:

\begin{align} \tanh(-x) &= -\tanh (x) \\ \coth(-x) &= -\coth (x) \\ \operatorname{sech}(-x) &= \operatorname{sech}(x) \\ \operatorname{csch}(-x) &= -\operatorname{csch}(x) \end{align}

It can be seen that cosh x and sech x are even functions; the others are odd functions.

\begin{align} \operatorname{arsech}(x) &= \operatorname{arcosh} \left(\frac{1}{x}\right) \\ \operatorname{arcsch}(x) &= \operatorname{arsinh} \left(\frac{1}{x}\right) \\ \operatorname{arcoth}(x) &= \operatorname{artanh} \left(\frac{1}{x}\right) \end{align}

Hyperbolic sine and cosine satisfy the identity

which is similar to the Pythagorean trigonometric identity. One also has

\begin{align} \operatorname{sech} ^{2}(x) &= 1 - \tanh^{2}(x) \\ \coth^{2}(x) &= 1 + \operatorname{csch}^{2}(x) \end{align}

for the other functions.

The hyperbolic tangent is the solution to the nonlinear boundary value problem:

It can be shown that the area under the curve of cosh (x) is always equal to the arc length:

Sums of arguments:

\begin{align} \cosh {(x + y)} &= \sinh{(x)} \cdot \sinh {(y)} + \cosh {(x)} \cdot \cosh {(y)} \\ \sinh {(x + y)} &= \cosh{(x)} \cdot \sinh {(y)} + \sinh {(x)} \cdot \cosh {(y)} \end{align}

Sum and difference of cosh and sinh:

\begin{align} \cosh{(x)} + \sinh{(x)} &= e^x \\ \cosh{(x)} - \sinh{(x)} &= e^{-x} \end{align}

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