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The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument , either circular angle or hyperbolic angle . Since the area of a circular sector with radius r and angle u (in radians) is r 2 u /2 , it will be equal to u when r = √ 2 .
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
The spherical triangle identities are written in terms of the ordinary trigonometric functions but differ from the plane triangle identities. Hyperbolic trigonometry: Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions. Hyperbolic functions in Euclidean geometry: The unit circle is parameterized by (cos t, sin t ...
Pages in category "Hyperbolic functions" The following 25 pages are in this category, out of 25 total. This list may not reflect recent changes. ...
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
In mathematics, hyperbolic trigonometry can mean: The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles in plane geometry) The use of the hyperbolic functions; The use of gyrotrigonometry in hyperbolic geometry
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