Swap the circle for a hyperbola and trigonometry gets a twin
Plot cosine against sine and you trace a unit circle; plot cosh against sinh and you trace the right half of a unit hyperbola. These hyperbolic functions mimic trigonometry closely, turn up in hanging chains and relativity, and were first named by an Italian mathematician in 1757.
The parallels run deep. Where sine differentiates to cosine and cosine to minus sine, sinh and cosh simply differentiate to each other, with no sign change. Their input is a hyperbolic angle, whose size equals the area of a sector bounded by the curve xy = 1, just as an ordinary angle can be read from a circular sector. In exponential form, sinh x is half the difference of e to the x and e to the minus x, the odd part of the exponential, and cosh x is half their sum, the even part. Tangent, cotangent, secant and cosecant each have hyperbolic versions built from these two, along with inverses written arsinh, arcosh and so on.
Complex numbers tie the two families together. Feed an imaginary angle to ordinary sine or cosine and hyperbolic functions come out. Sinh and cosh are entire, defined smoothly across the whole complex plane, so the others are meromorphic there, and the Lindemann-Weierstrass theorem shows they give transcendental values at every nonzero algebraic input.
Applications are wide. The shape of a hanging chain, the catenary, solves an equation whose answer involves cosh. The functions appear in solutions of cubic equations and of Laplace's equation, which governs problems in electromagnetism, heat flow and fluids. In hyperbolic geometry they express the angle of parallelism, and in special relativity a Lorentz boost becomes a rotation through a hyperbolic angle.
The history is gradual. Gerardus Mercator's map projection of around 1566 required tables solving a transcendental equation now written with hyperbolic functions, the earliest known such calculation. Newton noted a likeness between circular and hyperbolic sectors in the Principia of 1687, and Roger Cotes proposed using the imaginary unit to turn a prolate spheroid into an oblate one. Vincenzo Riccati formally introduced the functions in 1757 with the abbreviations Sh. and Ch., Daviet de Foncenex extended de Moivre's formula to them by 1759, and in the 1760s Johann Heinrich Lambert organised the subject, credited Riccati with the names and gave the modern abbreviations.
Source: Hyperbolic functions
