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In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.
A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter. Every diameter is a chord, but not every chord is a diameter.
In group theory, a branch of mathematics, the term order is used in two unrelated senses:
The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
In mathematics, Lagrange's theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange:
Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī was a mathematician, Islamic scholar, and Qadi from Al-Andalus. Al-Jayyānī wrote important commentaries on Euclid's Elements and he wrote the first known treatise on spherical trigonometry.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
Trigonometry is a branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics.
Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, hence it was adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus.
The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. In particular, 3rd-century astronomers first noted that the ratio of the lengths of two sides of a right-angled triangle depends only of one acute angles of the triangle. These dependencies are now called trigonometric functions.
Ordinary trigonometry studies triangles in the Euclidean plane R2. There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers: right-angled triangle definitions, unit-circle definitions, series definitions, definitions via differential equations, definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.
In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of L. Any Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian L is called degenerate if the Euler–Lagrange operator of L satisfies non-trivial Noether identities. In this case Euler–Lagrange equations are not independent.
de Moivre's theorem may be:
In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:
