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chmat
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There are 5 letters in CHMAT ( A1C3H4M3T1 )
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In mathematics, Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. * The basic hyperbolic functions are:* hyperbolic sine "sinh" (), * hyperbolic cosine "cosh" (),from which are derived: * * hyperbolic tangent "tanh" (), * hyperbolic cosecant "csch" or "cosech" ( or ) * hyperbolic secant "sech" (), * hyperbolic cotangent "coth" (),corresponding to the derived trigonometric functions. * The inverse hyperbolic functions are: * * area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") * and so on. * 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 equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. * Hyperbolic functions occur in the solutions of many linear differential equations (for example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. * In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. * By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. The abbreviations sh, ch, th, cth are also at disposition, their use depending more on personal preference of mathematics of influence than on the local language. |