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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called
The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is loge x. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. A hyperbolic sector is a region of the Cartesian plane {( x, y)} bounded by rays from the origin to two points ( a, 1/ a) and (
Note that unlike circular angle, hyperbolic angle is unbounded, as is the function loge x, a fact related to the unbounded nature of the harmonic series. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series The hyperbolic angle is considered to be negative when 0 < x < 1.
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. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions Dependent variables and independent variables refer to values that change in relationship to each other In Mathematics, the term hyperbolic triangle has more than one meaning Thus this parameter becomes one of the most useful in the calculus of a real variable. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, the real numbers may be described informally in several different ways
The quadrature of the hyperbola is the evaluation of the area swept out by a radial segment from the origin as the terminus moves along the hyperbola, just the topic of hyperbolic angle. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension The quadrature of the hyperbola was first accomplished by Gregoire de Saint-Vincent in 1647 in his momentous Opus geometricum quadrature circuli et sectionum coni. Grégoire de Saint-Vincent ( March 22 1584 Bruges – June 5 1667 Ghent) a Jesuit, was a Mathematician As David Eugene Smith wrote in 1925:
- [He made the] quadrature of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series the abscissas increased in geometric series. David Eugene Smith PhD LLD (1860&ndash1944 was an American mathematician and educator
- History of Mathematics, pp. 424,5 v. 1
The upshot was the logarithm function, as now understood as the area under y = 1/x to the right of x = 1. As an example of a transcendental function, the logarithm is more familiar than its motivator, the hyperbolic angle. A transcendental function is a function that does not satisfy a Polynomial equation whose Coefficients are themselves polynomials in contrast to an Nevertheless, the hyperbolic angle plays a role when the theorem of Saint-Vincent is advanced with squeeze mapping. In Linear algebra, a squeeze mapping is a type of Linear map that preserves Euclidean Area of regions in the Cartesian plane, but is not a
When Ludwik Silberstein penned his popular textbook on the new theory of relativity, he used the rapidity concept based on hyperbolic angle a where tanh a = v/c, the ratio of velocity v to the speed of light. He wrote:
- It seems worth mentioning that to unit rapidity corresponds a huge velocity, amounting to 3/4 of the velocity of light; more accurately we have v = (. 7616) c for a = 1.
- . . . the rapidity a = 1, . . . consequently will represent the velocity . 76 c which is a litle above the velocity of light in water.
Silberstein also uses Lobachevsky's concept of angle of parallelism Π(a) to obtain cos Π(a) = v/c. Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N In Hyperbolic geometry, the angle of parallelism Φ is the Angle at one vertex of a right Hyperbolic triangle that has two asymptotic parallel sides
References
- John Stillwell (1998) Numbers and Geometry exercise 9. 5. 3, p. 298, Springer-Verlag ISBN 0-387-98289-2.
- Ludwik Silberstein (1914) Theory of Relativity, Cambridge University Press, pp. Ludwik Silberstein (1872 – 1948 was a Polish-American physicist that helped make Special relativity and General relativity staples of university coursework Optics Book of Optics 180-1.
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