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Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified.
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
A left identity element that is also a right identity element if called an identity element. The empty set ∅ {\displaystyle \varnothing } is an identity element of binary union ∪ {\displaystyle \cup } and symmetric difference , {\displaystyle \triangle ,} and it is also a right identity element of set subtraction ∖ : {\displaystyle ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
The image of the function is the set of all output values it may produce, that is, the image of . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion is rarely used.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements.
The octant of a sphere is a spherical triangle with three right angles.. Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
The name "trisecant identity" refers to the geometric interpretation given by Mumford (1984, p.3.219), who used it to show that the Kummer variety of a genus g Riemann surface, given by the image of the map from the Jacobian to projective space of dimension induced by theta functions of order 2, has a 4-dimensional space of trisecants.