Trigonometry |
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In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [1] [2]
A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel in 1746. Thomas Simpson published the now-standard expression in 1748. Karl Mollweide republished the same result in 1808 without citing those predecessors. [3]
It can be used to check the consistency of solutions of triangles. [4]
Let and be the lengths of the three sides of a triangle. Let and be the measures of the angles opposite those three sides respectively. Mollweide's formulas are
Because in a planar triangle these identities can alternately be written in a form in which they are more clearly a limiting case of Napier's analogies for spherical triangles (this was the form used by Von Oppel),
Dividing one by the other to eliminate results in the law of tangents,
In terms of half-angle tangents alone, Mollweide's formula can be written as
or equivalently
Multiplying the respective sides of these identities gives one half-angle tangent in terms of the three sides,
which becomes the law of cotangents after taking the square root,
where is the semiperimeter.
The identities can also be proven equivalent to the law of sines and law of cosines.
In spherical trigonometry, the law of cosines and derived identities such as Napier's analogies have precise duals swapping central angles measuring the sides and dihedral angles at the vertices. In the infinitesimal limit, the law of cosines for sides reduces to the planar law of cosines and two of Napier's analogies reduce to Mollweide's formulas above. But the law of cosines for angles degenerates to By dividing squared side length by the spherical excess we obtain a non-vanishing ratio, the spherical trigonometry relation:
In the infinitesimal limit, as the half-angle tangents of spherical sides reduce to lengths of planar sides, the half-angle tangent of spherical excess reduces to twice the area of a planar triangle, so on the plane this is:
and likewise for and
As corollaries (multiplying or dividing the above formula in terms of and ) we obtain two dual statements to Mollweide's formulas. The first expresses the area in terms of two sides and the included angle, and the other is the law of sines:
We can alternately express the second formula in a form closer to one of Mollweide's formulas (again the law of tangents):
A generalization of Mollweide's formula holds for a cyclic quadrilateral Denote the lengths of sides and and angle measures and If is the point of intersection of the diagonals, denote Then: [5]
Several variant formulas can be constructed by substituting based on the cyclic quadrilateral identities,
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and the formulas above simplify to the analogous triangle formulas.
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In geometry, Heron's formula gives the area of a triangle in terms of the three side lengths a, b, c. Letting be the semiperimeter of the triangle, the area A is
In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle through the point at angle onto the line through the angles . Among these formulas are the following:
In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all the trisectors are intersected, one obtains four other equilateral triangles.
In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not.
There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German.
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya.
Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and , the law of cosines states:
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
In trigonometry, Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points A, B, and two unknown points P1, P2. From P1 and P2 an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of P1 and P2. See figure; the angles measured are (α1, β1, α2, β2).
In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.
Pentagramma mirificum is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici Logarithmorum Canonis Descriptio along with rules that link the values of trigonometric functions of five parts of a right spherical triangle. The properties of pentagramma mirificum were studied, among others, by Carl Friedrich Gauss.