三角与双曲三角及其反函数的转换

\begin{matrix} \begin{array}{lll} \cos x = \frac{e^{i x} + e^{- i x}}{2}, & \sin x = \frac{e^{i x} - e^{- i x}}{2 i}, & \tan x = \frac{1}{i} \frac{e^{i x} - e^{- i x}}{e^{i x} + e^{- i x}}, \\ \cosh x = \frac{e^{x} + e^{- x}}{2}, & \sinh x = \frac{e^{x} - e^{- x}}{2}, & \tanh x = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} . \\ \cos i x = \cosh x, & \sin i x = i \sinh x, & \tan i x = i \tanh x, \\ \cosh i x = \cos x, & \sinh i x = i \sin x, & \tanh i x = i \tan x . \end{array} \\ \begin{array}{lll} \arccos x = i \ln (\sqrt{1 - x^{2}} - i x), & \arcsin x = i \ln (\sqrt{1 - x^{2}} + i x), & \arctan x = \frac{1}{2 i} \ln \frac{1 + i x}{1 - i x} . \\ arch x = \ln (x + \sqrt{x^{2} - 1}), & arsh x = \ln (x + \sqrt{x^{2} + 1}), & arth x = \frac{1}{2} \ln \frac{1 + x}{1 - x} . \\ \arccos x = i arch x, & \arcsin i x = i arsh x, & \arctan i x = i arth x, \\ arch x = i \arccos x, & arsh i x = i \arcsin x, & arth i x = i \arctan x . \end{array} \end{matrix}