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What (who) is elliptic cosine - definition
![Plot of the degenerate Jacobi curve (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' = ∞) and the twelve Jacobi Elliptic functions pq(''u'',1) for a particular value of angle ''φ''. The solid curve is the degenerate ellipse (''x''<sup>2</sup> = 1) with ''m'' = 1 and ''u'' = ''F''(''φ'',1) where ''F''(·,\middot') is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for ''m'' = 0 (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.](https://commons.wikimedia.org/wiki/Special:FilePath/JacobiElliptic.HT.svg?width=200 "Plot of the degenerate Jacobi curve (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' = ∞) and the twelve Jacobi Elliptic functions pq(''u'',1) for a particular value of angle ''φ''. The solid curve is the degenerate ellipse (''x''<sup>2</sup> = 1) with ''m'' = 1 and ''u'' = ''F''(''φ'',1) where ''F''(·,\middot') is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for ''m'' = 0 (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.")
![Plot of the Jacobi ellipse (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' real) and the twelve Jacobi elliptic functions ''pq''(''u'',''m'') for particular values of angle ''φ'' and parameter ''b''. The solid curve is the ellipse, with ''m'' = 1 − 1/''b''<sup>2</sup> and ''u'' = ''F''(''φ'',''m'') where ''F''(·,·) is the [[elliptic integral]] of the first kind (with parameter <math>m=k^2</math>). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at ''x'' = cd crossing the ''x''-axis at dc are shown in light grey.](https://commons.wikimedia.org/wiki/Special:FilePath/Jacobi Elliptic Functions (on Jacobi Ellipse).svg?width=200 "Plot of the Jacobi ellipse (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' real) and the twelve Jacobi elliptic functions ''pq''(''u'',''m'') for particular values of angle ''φ'' and parameter ''b''. The solid curve is the ellipse, with ''m'' = 1 − 1/''b''<sup>2</sup> and ''u'' = ''F''(''φ'',''m'') where ''F''(·,·) is the [[elliptic integral]] of the first kind (with parameter <math>m=k^2</math>). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at ''x'' = cd crossing the ''x''-axis at dc are shown in light grey.")
![Plot of the Jacobi hyperbola (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' imaginary) and the twelve Jacobi Elliptic functions pq(''u'',''m'') for particular values of angle ''φ'' and parameter ''b''. The solid curve is the hyperbola, with ''m'' = 1 − 1/''b''<sup>2</sup> and ''u'' = ''F''(''φ'',''m'') where ''F''(·,·) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, ''σ'' = sin(''φ'')cos(''φ'').](https://commons.wikimedia.org/wiki/Special:FilePath/Jacobi Elliptic Functions (on Jacobi Hyperbola).svg?width=200 "Plot of the Jacobi hyperbola (''x''<sup>2</sup> + ''y''<sup>2</sup>/''b''<sup>2</sup> = 1, ''b'' imaginary) and the twelve Jacobi Elliptic functions pq(''u'',''m'') for particular values of angle ''φ'' and parameter ''b''. The solid curve is the hyperbola, with ''m'' = 1 − 1/''b''<sup>2</sup> and ''u'' = ''F''(''φ'',''m'') where ''F''(·,·) is the [[elliptic integral]] of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, ''σ'' = sin(''φ'')cos(''φ'').")
 durch eine Fläche -Schilling V, 1 - 317-.jpg?width=200 "Model of the Jacobi amplitude (measured along vertical axis) as a function of independent variables ''u'' and the modulus ''k''")
![Ottoman Turkey]] with axes for looking up the sine and [[versine]] of angles](https://commons.wikimedia.org/wiki/Special:FilePath/Khalili Collection Islamic Art sci 0040.1 CROP.jpg?width=200 "Ottoman Turkey]] with axes for looking up the sine and [[versine]] of angles")
![The quadrants of the unit circle and of sin(''x''), using the [[Cartesian coordinate system]]](https://commons.wikimedia.org/wiki/Special:FilePath/Sine quads 01 Pengo.svg?width=200 "The quadrants of the unit circle and of sin(''x''), using the [[Cartesian coordinate system]]")
Wikipedia
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.
