trigonometric
we use the familiar variable \(z = x + \href{https://en.wikipedia.org/wiki/Complex_number}{\mathrm i}y\quad x, y\in\href{https://en.wikipedia.org/wiki/Real_number}{\mathbb R}\)
\(\href{https://en.wikipedia.org/wiki/Sine}{\sin}(z) = \sin(x)\cosh(y) + \mathrm i\cos(x)\sinh(y)\)
\(\href{https://en.wikipedia.org/wiki/Cosine}{\cos}(z) = \cos(x)\cosh(y) - \mathrm i\sin(x)\sinh(y)\)
\(\href{https://en.wikipedia.org/wiki/Tangent_(trigonometry)}{\tan}(z) = \dfrac{\mathrm e^{+\mathrm iz} - \mathrm e^{-\mathrm iz}}{\mathrm e^{+\mathrm iz} + \mathrm e^{-\mathrm iz}}\dfrac{1}{\mathrm i}\)
\(\href{https://en.wikipedia.org/wiki/Cotangent}{\cot}(z) = \dfrac{\mathrm e^{+\mathrm iz} + \mathrm e^{-\mathrm iz}}{\mathrm e^{+\mathrm iz} - \mathrm e^{-\mathrm iz}}\dfrac{\mathrm i}{1}\)
\(\href{https://en.wikipedia.org/wiki/Secant_(trigonometry)}{\sec}(z) = \dfrac{1}{\cos(z)}\)
\(\href{https://en.wikipedia.org/wiki/Cosecant}{\csc}(z) = \dfrac{1}{\sin(z)}\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{asin}}(z) = -\mathrm i \ln\left(\sqrt{1 - z^2} + \mathrm iz\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{acos}}(z) = -2\mathrm i\ln\left(\sqrt{\dfrac{1+z}{2}}+\mathrm i\sqrt{\dfrac{1-z}{2}}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{atan}}(z) = \dfrac{\mathrm i}{2}\ln\left(\dfrac{\mathrm i + \mathrm z}{\mathrm i - \mathrm z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{acot}}(z) = \operatorname{atan}\left(\dfrac{1}{z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{asec}}(z) = \operatorname{acos}\left(\dfrac{1}{z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{\operatorname{acsc}}(z) = \operatorname{asin}\left(\dfrac{1}{z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\operatorname{sinh}}(z) = \sinh(x)\cos(y) + \mathrm i\cosh(x)\sin(y)\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\operatorname{cosh}}(z) = \cosh(x)\cos(y) + \mathrm i\sinh(x)\sin(y)\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\tanh}(z) = -\mathrm i\tanh(\mathrm i z)\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\coth}(z) = \dfrac{\mathrm e^{+\mathrm jz} + \mathrm e^{−\mathrm jz}}{\mathrm e^{+\mathrm jz} - \mathrm e^{−\mathrm jz}}\dfrac{\mathrm j}{1}\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\operatorname{sech}}(z) = \dfrac{1}{\cosh(z)}\)
\(\href{https://en.wikipedia.org/wiki/Hyperbolic_functions}{\operatorname{csch}}(z) = \dfrac{1}{\sinh(z)}\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{asinh}}(z) = \ln\left(\sqrt{z^2 + 1} + z\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{acosh}}(z) = 2\ln\left(\sqrt{\dfrac{z + 1}{2}} + \sqrt{\dfrac{z - 1}{2}}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{atanh}}(z) = \dfrac{1}{2}\ln\left(\dfrac{1+z}{1-z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{acoth}}(z) = \operatorname{atanh}\left(\dfrac{1}{z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{asech}}(z) = \operatorname{acosh}\left(\dfrac{1}{z}\right)\)
\(\href{https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions}{\operatorname{acsch}}(z) = \operatorname{asinh}\left(\dfrac{1}{z}\right)\)
the 12 parabolic trig functions are not included because they are trivial: \(\operatorname{sinp}(z) = z,\quad \operatorname{cosp}(z) = 1\)
when branch values differ by approach direction, the input datatype must encode that direction; otherwise, evaluation raises DatatypeError.