쌍곡함수

\(\sinh x = \frac{e^x - e^{-x}}{2}\)

\(\cosh x = \frac{e^{x} + e^{-x}}{2}\)

\(\tanh x = \frac{\sinh x}{\cosh x} = \frac {\frac{1}{2}(e^x - e^{-x})} {\frac{1}{2}(e^x + e^{-x})} = \frac{e^{2x} - 1} {e^{2x} + 1}\)

\(\operatorname{sech}\,x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}\)

\(\cosh^2 x - \sinh^2 x = 1\)

\(\tanh ^{2}x=1-\operatorname{sech}^{2}x\)

\((\sinh x)' = \frac{e^x + e^{-x}}{2}=\cosh x\)

\((\cosh x)' = \frac{e^{x} - e^{-x}}{2}=\sinh x\)

\((\tanh x)' = \frac{\cosh^2 x- \sinh^2 x}{\cosh^2 x}=\operatorname{sech}^{2}x\)

\(\cosh \left(\theta _1\right) \cosh \left(\theta _2\right)-\sinh \left(\theta _1\right) \sinh \left(\theta _2\right)=\cosh \left(\theta _1-\theta _2\right)\)

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