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Hyperbolic functions - Wikipedia
https://en.wikipedia.org/wiki/Hyperbolic_functions
WebIn 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.
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Hyperbolic Functions - Math is Fun
https://www.mathsisfun.com/sets/function-hyperbolic.html
WebFrom sinh and cosh we can create: Hyperbolic tangent "tanh" (pronounced "than"): tanh(x) = sinh(x) cosh(x) = e x − e-x e x + e-x. tanh vs tan . Hyperbolic cotangent: coth(x) = cosh(x) sinh(x) = e x + e-x e x − e-x . Hyperbolic secant: sech(x) = 1 cosh(x) = 2 e x + e-x . Hyperbolic cosecant "csch" or "cosech": csch(x) = 1 sinh(x) = 2 e x − ...
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Hyperbolic Functions - sinh, cosh, tanh, coth, sech, csch - Math10
https://www.math10.com/en/algebra/hyperbolic-functions/hyperbolic-functions.html
WebDefinition of hyperbolic functions. Hyperbolic sine of x. \displaystyle \text {sinh}\ x = \frac {e^ {x} - e^ {-x}} {2} sinh x = 2ex −e−x. Hyperbolic cosine of x. \displaystyle \text {cosh}\ x = \frac {e^x + e^ {-x}} {2} cosh x = 2ex +e−x. Hyperbolic tangent of x.
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Hyperbolic Trigonometric Functions | Brilliant Math & Science Wiki
https://brilliant.org/wiki/hyperbolic-trigonometric-functions/
WebThe hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle (x = \cos t (x = cost and y = \sin t) y = sint) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations: x = \cosh a = \dfrac {e^a + e^ {-a}} {2},\quad y = \sinh a = \dfrac {e^a - e^ {-a}} {2 ...
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Introduction to the Hyperbolic Sine Function - Wolfram
https://functions.wolfram.com/ElementaryFunctions/Sinh/introductions/Sinh/ShowAll.html
WebDefining the hyperbolic sine function. The hyperbolic sine function is an old mathematical function. It was first used in the works of V. Riccati (1757), D. Foncenex (1759), and J. H. Lambert (1768). The hyperbolic sine function is easily defined as the half difference of two exponential functions in the points and :
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Sinh Calculator
https://www.omnicalculator.com/math/sinh
WebJan 18, 2024 · sinh is an odd function, i.e., sinh ( − x) = − sinh ( x) \sinh (-x) = -\sinh (x) sinh(−x) = −sinh(x); sinh is increasing; sinh ( 0) = 0. \sinh (0) = 0 sinh(0) = 0; sinh is not periodic; sinh is not bounded; and. sinh is a bijection, so it has an inverse.
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Hyperbolic Sine -- from Wolfram MathWorld
https://mathworld.wolfram.com/HyperbolicSine.html
WebMar 15, 2024 · The hyperbolic sine is defined as sinhz=1/2(e^z-e^(-z)). (1) The notation shz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Sinh[z]. Special values include sinh0 = 0 (2) sinh(lnphi) = 1/2, (3) where phi is the golden ratio. The value sinh1=1.17520119...
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Sinh Definition (Illustrated Mathematics Dictionary) - Math is Fun
https://www.mathsisfun.com/definitions/sinh.html
Websinh (x) = (e x − e −x) / 2. Pronounced "shine". See: Hyperbolic Functions. Hyperbolic Functions. Illustrated definition of Sinh: The Hyperbolic Sine Function sinh (x) (esupxsup …
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Hyperbolic Trigonomic Identities - Math2.org
http://math2.org/math/trig/hyperbolics.htm
WebHyperbolic Definitions. sinh (x) = ( e x - e -x )/2. csch (x) = 1/sinh (x) = 2/ ( e x - e -x ) cosh (x) = ( e x + e -x )/2. sech (x) = 1/cosh (x) = 2/ ( e x + e -x ) tanh (x) = sinh (x)/cosh (x) = ( e x - e -x )/ ( e x + e -x ) coth (x) = 1/tanh (x) = ( e x + e -x )/ ( e x - e -x ) cosh 2 (x) - sinh 2 (x) = 1. tanh 2 (x) + sech 2 (x) = 1.
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Trigonometry/Cosh, Sinh and Tanh - Wikibooks
https://en.wikibooks.org/wiki/Trigonometry/Cosh,_Sinh_and_Tanh
WebSep 25, 2020 · Equivalently, Reciprocal functions may be defined in the obvious way: 1 - tanh 2 (x) = sech 2 (x); coth 2 (x) - 1 = cosech 2 (x) It is easily shown that , analogous to the result In consequence, sinh (x) is always less in absolute value than cosh (x). sinh (-x) = -sinh (x); cosh (-x) = cosh (x); tanh (-x) = -tanh (x).
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