Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and sinh(t). Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: hyperbolic sine "sinh" (), hyperbolic cosine "cosh" (), from which are derived: hyperbolic tangent "tanh" (), hyperbolic cotangent "coth" (), hyperbolic secant "sech" (), hyperbolic cosecant "csch" or "cosech" () corresponding to the derived trigonometric functions. The inverse hyperbolic functions are: inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") inverse hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh") inverse hyperbolic tangent "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh") inverse hyperbolic cotangent "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth") inverse hyperbolic secant "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech") inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech") The hyperbolic functions take an argument called a hyperbolic angle. The magnitude of a hyperbolic angle is the area of its hyperbolic sector to xy = 1. The hyperbolic functions may be defined in terms of the legs of a right triangle...