Tangente

Tangente
Name	Tangente
Field	Mathematics
Introduced	Antiquity
Notation	tan, tg

Tangente is a mathematical function and geometric concept central to trigonometry, analytic geometry, and calculus. It relates angles in triangles and points on the unit circle to ratios of side lengths and coordinates, and it appears across physics, engineering, navigation, and signal processing. The concept has deep historical roots from ancient geometry through Renaissance analysis to modern computational applications.

Etymology

The term derives from Latin roots used in medieval mathematics and was popularized by Renaissance translators of Arabic scholarship. Related lexical developments appear alongside names associated with Euclid, Ptolemy, Al-Khwarizmi, and Fibonacci in the transmission of trigonometric vocabulary. Renaissance mathematicians such as Regiomontanus and Viète introduced notation and terminology that interacted with contemporaneous treatises by Copernicus and Cardano.

Mathematical definitions

In right triangle contexts the function equals the ratio of two side lengths; classical expositions by Euclid, Ptolemy, and Hipparchus influenced later tabulations in works by al-Battani and Nasir al-Din al-Tusi. In analytic geometry the function maps an angle on the unit circle to the quotient of the sine and cosine coordinates, a formulation employed in texts by Euler, Gauss, and Cauchy. Complex analysis extends the definition using exponential forms encountered in studies by Fourier, Riemann, and Weierstrass, and modern algebraic treatments reference results from Hilbert and Noether.

Geometric properties

On the unit circle the function corresponds to a line tangent to the circle at a point, a picture used in demonstrations by Archimedes and Apollonius; related constructions appear in treatises by Descartes and Newton. Periodicity, odd symmetry, and vertical asymptotes connect to classical investigations by Bernoulli and Lagrange. Identities involving angle addition and multiple angles were derived and refined in works by Napier, Euler, and Fourier; geometric proofs often reference constructions similar to those found in diagrams attributed to Kepler and Huygens.

Calculus and analysis

Differentiation and integration formulas for the function are standard in the teachings of Leibniz and Newton and were systematized in textbooks by Fourier, Cauchy, and Lebesgue. Series expansions, including Maclaurin and Taylor series, trace to contributions by Gregory, Mercator, and Taylor, while convergence analyses invoke results by Abel and Dirichlet. The function’s complex singularities and residues have been explored in the frameworks developed by Riemann, Hadamard, and Gelfand, and its role in ordinary differential equations appears in canonical works by Sturm and Liouville.

Applications

Practical uses appear in navigation manuals from Polynesian voyaging to maritime charts influenced by Mercator and Cook, and in surveying techniques propagated by Cassini and Struve. In optics and acoustics, analyses by Fresnel and Helmholtz employ the function in wavefront and harmonic models; electrical engineering and signal processing use it in phasor analysis and filter design following developments by Maxwell, Nyquist, and Shannon. Structural engineering and civil works reference formulations found in projects by Eiffel and Brunel, while aerospace guidance, satellite telemetry, and control theory apply trigonometric models developed in programs associated with von Kármán and NACA.

Historical development

Early tabulations of trigonometric ratios by Indian astronomers such as Āryabhata and Brahmagupta influenced Hellenistic work by Hipparchus and Ptolemy; these streams merged in Arabic scholarship exemplified by Al-Battani and Al-Khujandi. Medieval transmissions via Toledo and Palermo brought refinements to Europe, where scholars like Regiomontanus, Rheticus, and Stifel produced systematic treatises. The formal analytic approach emerged with Descartes and Fermat, matured through Euler and Gauss, and integrated into modern curricula influenced by Hilbert, Bourbaki, and contemporary mathematical societies.

Cultural and linguistic uses

The concept’s terminology has penetrated vernaculars and technical jargons across Europe and Asia, appearing in works by Dante, Cervantes, Goethe, and Pushkin as metaphorical language in poetry and prose. Cartographic conventions and idioms in maritime lore reference the function indirectly via navigational practice chronicled by Magellan and Drake. In modern languages usage guides and dictionaries produced by institutions such as the Académie française and the Real Academia Española reflect standardized notations like tan and tg; computational toolchains from IBM and Bell Laboratories standardized symbolic and programming conventions. Category:Trigonometry