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- Hyperbolic_angle abstract "In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola.A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1.The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln x.Note that unlike circular angle, hyperbolic angle is unbounded, as is the function ln x, a fact related to the unbounded nature of the harmonic series.The hyperbolic angle in standard position is considered to be negative when 0 < x < 1.Suppose ab = 1 and cd = 1 with c > a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy = 1. Then the squeeze mapping with diagonal elements b and a maps this interval to the standard position hyperbolic angle that runs from (1, 1) to (bc, ad). By the result of Gregoire de Saint-Vincent, the hyperbolic sector determined by (a, b) and (c, d) has the same area as this standard position angle, and the magnitude of the hyperbolic angle is taken to be this area.The hyperbolic functions sinh, cosh, and tanh use the hyperbolic angle as their independent variable because their values may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle.Thus this parameter becomes one of the most useful in the calculus of a real variable.".
- Hyperbolic_angle thumbnail Cartesian_hyperbolic_triangle.svg?width=300.
- Hyperbolic_angle wikiPageExternalLink applicationofhyp00kennrich.
- Hyperbolic_angle wikiPageExternalLink e5.html.
- Hyperbolic_angle wikiPageID "1139981".
- Hyperbolic_angle wikiPageRevisionID "599209236".
- Hyperbolic_angle caption "The hyperbolic angle u is a real number that is the argument of the hyperbolic functions sinh and cosh. It determines a hyperbolic sector that has area u. The legs of the hyperbolic triangle are proportional to sinh and cosh.".
- Hyperbolic_angle caption "Top: Positive and negative hyperbolic angles. Bottom: The difference between the two positive angles is shown as Δu = u2 − u1.".
- Hyperbolic_angle hasPhotoCollection Hyperbolic_angle.
- Hyperbolic_angle image "Cartesian hyperbolic triangle.svg".
- Hyperbolic_angle image "Hyperbolic angle2.svg".
- Hyperbolic_angle width "190".
- Hyperbolic_angle subject Category:Angle.
- Hyperbolic_angle subject Category:Differential_calculus.
- Hyperbolic_angle subject Category:Integral_calculus.
- Hyperbolic_angle comment "In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle.".
- Hyperbolic_angle label "Hyperbolic angle".
- Hyperbolic_angle label "Hyperboolhoek".
- Hyperbolic_angle label "Ángulo hiperbólico".
- Hyperbolic_angle label "Ângulo hiperbólico".
- Hyperbolic_angle label "双曲角".
- Hyperbolic_angle sameAs Ángulo_hiperbólico.
- Hyperbolic_angle sameAs Hyperboolhoek.
- Hyperbolic_angle sameAs Ângulo_hiperbólico.
- Hyperbolic_angle sameAs m.049rzz.
- Hyperbolic_angle sameAs Q1003216.
- Hyperbolic_angle sameAs Q1003216.
- Hyperbolic_angle wasDerivedFrom Hyperbolic_angle?oldid=599209236.
- Hyperbolic_angle depiction Cartesian_hyperbolic_triangle.svg.
- Hyperbolic_angle isPrimaryTopicOf Hyperbolic_angle.