1.
Sign
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A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a relation to its object—for instance, thunder is a sign of storm. The physical objects most commonly referred to as signs generally inform or instruct using written text, symbols, the philosophical study of signs and symbols is called semiotics, this includes the study of semiosis, which is the way in which signs operate. Semiotics, epistemology, logic, and philosophy of language are concerned about the nature of signs, what they are, the nature of signs and symbols and significations, their definition, elements, and types, is mainly established by Aristotle, Augustine, and Aquinas. Facebook A sign can denote any of the following, Sun signs in astrology Sign or signing, in communication, communicating via hand gestures, gang signal Sign, in Tracking, also known as Spoor, trace evidence left on the ground after passage. Also, the sign of a permutation tells whether it is the product of an even or odd number of transpositions. Signedness, in computing, is the property that a representation of a number has one bit, the sign bit, a number is called signed if it contains a sign bit, otherwise unsigned. St. Augustine was the first man who synthesized the classical, for him a sign is a thing which is used to signify other things and to make them come to mind. The most common signs are spoken and written words, although God cannot be fully expressible, Augustine gave emphasis to the possibility of God’s communication with humans by signs in Scripture. Augustine endorsed and developed the classical and Hellenistic theories of signs, if we match DDC with this division, the first part belongs to DDC Book IV and the second part to DDC Books I-III. Augustine, although influenced by these theories, advanced his own theory of signs, with whose help one can infer the mind of God from the events. Books II and III of DDC enumerate all kinds of signs, augustine’s understanding of signs includes several hermeneutical presuppositions as important factors. First, the interpreter should proceed with humility, because only a person can grasp the truth of Scripture. Third, the heart of interpreter should be founded, rooted, the sign does not function as its own goal, but its purpose lies in its role as a signification. God gave signs as a means to reveal himself, Christians need to exercise hermeneutical principles in order to understand that divine revelation, even if the Scriptural text is obscure, it has meaningful benefits. For the obscure text prevents us from falling into pride, triggers our intelligence, tempers our faith in the history of revelation, when interpreting signs, the literal meaning should first be sought, and then the figurative meaning. Moreover, he introduces the seven rules of Tyconius the Donatist to interpret the meaning of the Bible. In order to apply Augustines hermeneutics of the sign appropriately in modern times, every division of theology must be involved, the dictionary definition of sign at Wiktionary
2.
Sign (mathematics)
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
3.
Even and odd functions
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In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in areas of mathematical analysis, especially the theory of power series. The concept of evenness or oddness is defined for functions whose domain and this includes additive groups, all rings, all fields, and all vector spaces. Thus, for example, a function of a real variable could be even or odd, as could a complex-valued function of a vector variable. The examples are real-valued functions of a variable, to illustrate the symmetry of their graphs. Let f be a function of a real variable. Then f is even if the equation holds for all x and -x in the domain of f, f = f. Geometrically speaking, the face of an even function is symmetric with respect to the y-axis. Examples of even functions are |x|, x2, x4, cos, cosh, again, let f be a real-valued function of a real variable. Then f is odd if the equation holds for all x and -x in the domain of f, − f = f. Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, examples of odd functions are x, x3, sin, sinh, erf, or any linear combination of these. A functions being odd or even does not imply differentiability, or even continuity, for example, the Dirichlet function is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist, there is only 1 odd even function, f=0. If a function is even and odd, it is equal to 0 everywhere it is defined, if a function is odd, the absolute value of that function is an even function. The sum of two functions is even, and any constant multiple of an even function is even. The sum of two odd functions is odd, and any constant multiple of an odd function is odd, the difference between two odd functions is odd. The difference between two functions is even. The sum of an even and odd function is neither even nor odd, the product of two even functions is an even function
4.
Domain of a function
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In mathematics, and more specifically in naive set theory, the domain of definition of a function is the set of input or argument values for which the function is defined. That is, the function provides an output or value for each member of the domain, conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. For instance, the domain of cosine is the set of all real numbers, if the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the X-axis. Given a function f, X→Y, the set X is the domain of f, in the expression f, x is the argument and f is the value. One can think of an argument as a member of the domain that is chosen as an input to the function, the image of f is the set of all values assumed by f for all possible x, this is the set. The image of f can be the set as the codomain or it can be a proper subset of it. It is, in general, smaller than the codomain, it is the whole codomain if, a well-defined function must map every element of its domain to an element of its codomain. For example, the function f defined by f =1 / x has no value for f, thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is defined on R\ or the gap is plugged by explicitly defining f. If we extend the definition of f to f = {1 / x x ≠00 x =0 then f is defined for all real numbers, any function can be restricted to a subset of its domain. The restriction of g, A → B to S, where S ⊆ A, is written g |S, S → B. The natural domain of a function is the set of values for which the function is defined, typically within the reals. For instance the natural domain of square root is the non-negative reals when considered as a real number function, when considering a natural domain, the set of possible values of the function is typically called its range. There are two meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i. e. a function from a subset X of X to Y. Most mathematicians, including recursion theorists, use the domain of f for the set X of all values x such that f is defined. But some, particularly category theorists, consider the domain to be X, in category theory one deals with morphisms instead of functions. Morphisms are arrows from one object to another, the domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly, for example, the notion of restricting a morphism to a subset of its domain must be modified
5.
Codomain
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In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the f, X → Y. The codomain is also referred to as the range but that term is ambiguous as it may also refer to the image. The set F is called the graph of the function, the set of all elements of the form f, where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain, thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements y in its codomain for which the equation f = y does not have a solution. An alternative definition of function by Bourbaki, namely as just a functional graph, for example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple. With such a definition functions do not have a codomain, although some still use it informally after introducing a function in the form f, X → Y. For a function f, R → R defined by f, x ↦ x 2, or equivalently f = x 2, the codomain of f is R, but f does not map to any negative number. Thus the image of f is the set R0 +, i. e. the interval [0, an alternative function g is defined thus, g, R → R0 + g, x ↦ x 2. While f and g map a given x to the number, they are not, in this view. A third function h can be defined to demonstrate why, h, x ↦ x, the domain of h must be defined to be R0 +, h, R0 + → R. The compositions are denoted h ∘ f, h ∘ g, on inspection, h ∘ f is not useful. The codomain affects whether a function is a surjection, in that the function is surjective if, in the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection, each matrix represents a map with the domain R2 and codomain R2. Some transformations may have image equal to the codomain but many do not. Take for example the matrix T given by T = which represents a linear transformation that maps the point to, the point is not in the image of T, but is still in the codomain since linear transformations from R2 to R2 are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set, examining the differences between the image and codomain can often be useful for discovering properties of the function in question
6.
Period (mathematics)
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In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P
7.
Y-intercept
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As such, these points satisfy x =0. If the curve in question is given as y = f, functions which are undefined at x =0 have no y-intercept. If the function is linear and is expressed in slope-intercept form as f = a + b x, some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one y-intercept. Because functions associate x values to no more than one y value as part of their definition, analogously, an x-intercept is a point where the graph of a function or relation intersects with the x-axis. As such, these points satisfy y=0, the zeros, or roots, of such a function or relation are the x-coordinates of these x-intercepts. Unlike y-intercepts, functions of the form y = f may contain multiple x-intercepts, the x-intercepts of functions, if any exist, are often more difficult to locate than the y-intercept, as finding the y intercept involves simply evaluating the function at x=0. The notion may be extended for 3-dimensional space and higher dimensions, as well as for other coordinate axes, for example, one may speak of the I-intercept of the current-voltage characteristic of, say, a diode
8.
Zero of a function
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In other words, a zero of a function is an input value that produces an output of zero. A root of a polynomial is a zero of the polynomial function. If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis, an alternative name for such a point in this context is an x-intercept. Every equation in the unknown x may be rewritten as f =0 by regrouping all terms in the left-hand side and it follows that the solutions of such an equation are exactly the zeros of the function f. Every real polynomial of odd degree has an odd number of roots, likewise. Consequently, real odd polynomials must have at least one real root, the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs, vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Computing roots of functions, for polynomial functions, frequently requires the use of specialised or approximation techniques. However, some functions, including all those of degree no greater than 4. In topology and other areas of mathematics, the set of a real-valued function f, X → R is the subset f −1 of X. Zero sets are important in many areas of mathematics. One area of importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k, one defines the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. Then a subset V of An is called an algebraic set if V = Z for some S. These affine algebraic sets are the building blocks of algebraic geometry. Zero Pole Fundamental theorem of algebra Newtons method Sendovs conjecture Mardens theorem Vanish at infinity Zero crossing Weisstein, Eric W. Root
9.
Critical point (mathematics)
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In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0. Some authors include in the points the limit points where the function may be prolongated by continuity. For a differentiable function of real variables, a critical point is a value in its domain where all partial derivatives are zero. The value of the function at a point is a critical value. The interest of this lies in the fact that the points where the function has local extrema are critical points. This definition extends to differentiable maps between Rm and Rn, a point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points, in other words, the critical points are those where the implicit function theorem does not apply. The notion of a critical point allows the description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere and this occurs because of a critical point of the projection of the orbit into the ecliptic circle. A critical point or stationary point of a function of a single real variable. A critical value is the image f of a critical point. These concepts may be visualized through the graph of f, at a point, the graph has a horizontal tangent. Although it is easily visualized on the graph, the notion of point of a function must not be confused with the notion of critical point, in some direction. If g is a function of two variables, then g =0 is the implicit equation of a curve. A critical point of such a curve, for the parallel to the y-axis, is a point of the curve where ∂ g ∂ y =0. This means that the tangent of the curve is parallel to the y-axis, if is such a critical point, then x0 is the corresponding critical value. Such a critical point is called a bifurcation point, as, generally
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Inflection point
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In differential calculus, an inflection point, point of inflection, flex, or inflection is a point on a curve at which the curve changes from being concave to convex, or vice versa. A point where the curvature vanishes but does not change sign is called a point of undulation or undulation point. Inflection points are the points of the curve where the curvature changes its sign while a tangent exists, a differentiable function has an inflection point at if and only if its first derivative, f′, has an isolated extremum at x. That is, in some neighborhood, x is the one, if all extrema of f′ are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve. A rising point of inflection is a point where the derivative has a local minimum. For an algebraic curve, a non singular point is a point if and only if the multiplicity of the intersection of the tangent line. For a curve given by parametric equations, a point is a point if its signed curvature changes from plus to minus or from minus to plus. For a twice differentiable function, a point is a point on the graph at which the second derivative has an isolated zero. If x is a point for f then the second derivative, f″, is equal to zero if it exists. One also needs the lowest-order non-zero derivative to be of odd order, if the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in geometry, both inflection points and undulation points are usually called inflection points. An example of such a point is x =0 for the function f given by f = x4. If this is positive, the point is a point of inflection, if it is negative. 2) Another sufficient existence condition requires f′′ and f′′ to have signs in the neighborhood of x. Points of inflection can also be categorized according to whether f′ is zero or not zero. If f′ is zero, the point is a point of inflection if f′ is not zero. The tangent is the x-axis, which cuts the graph at this point, a non-stationary point of inflection can be visualised if the graph y = x3 is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero, some functions change concavity without having points of inflection
11.
Fixed point (mathematics)
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In mathematics, a fixed point of a function is an element of the functions domain that is mapped to itself by the function. That is to say, c is a point of the function f if. This means f = fn = c, an important terminating consideration when recursively computing f, a set of fixed points is sometimes called a fixed set. For example, if f is defined on the numbers by f = x 2 −3 x +4, then 2 is a fixed point of f. In graphical terms, a point means the point is on the line y = x. Points that come back to the value after a finite number of iterations of the function are called periodic points. A fixed point is a point with period equal to one. In projective geometry, a point of a projectivity has been called a double point. In Galois theory, the set of the points of a set of field automorphisms is a field called the fixed field of the set of automorphisms. An expression of prerequisites and proof of the existence of solution is given by the Banach fixed-point theorem. The natural cosine function has exactly one fixed point, which is attractive, in this case, close enough is not a stringent criterion at all—to demonstrate this, start with any real number and repeatedly press the cos key on a calculator. It eventually converges to about 0.739085133, which is a fixed point and that is where the graph of the cosine function intersects the line y = x. Not all fixed points are attractive, for example, x =0 is a fixed point of the function f = 2x, but iteration of this function for any value other than zero rapidly diverges. However, if the function f is differentiable in an open neighbourhood of a fixed point x0. Attractive fixed points are a case of a wider mathematical concept of attractors. An attractive fixed point is said to be a fixed point if it is also Lyapunov stable. A fixed point is said to be a stable fixed point if it is Lyapunov stable. The center of a linear differential equation of the second order is an example of a neutrally stable fixed point
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Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
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Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
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Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
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Outline of trigonometry
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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, geometry – mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is used extensively in trigonometry, angle – angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. D, ebook version, in PDF format, full text presented. Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company,1914, Trigonometry FAQ Trigonometry on Mathwords. com index of trigonometry entries on Mathwords. com Trigonometry on PlainMath. net Trigonometry Articles from PlainMath. Net
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History of trigonometry
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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 continued in Islamic mathematics. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics, the term trigonometry was derived from Greek τρίγωνον trigōnon, triangle and μέτρον metron, measure. Our modern word sine is derived from the Latin word sinus, the Arabic term is in origin a corruption of Sanskrit jīvā, or chord. Sanskrit jīvā in learned usage was a synonym of jyā chord, Sanskrit jīvā was loaned into Arabic as jiba. Particularly Fibonaccis sinus rectus arcus proved influential in establishing the term sinus, the words minute and second are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to first small parts and second small parts, the ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, based on one interpretation of the Plimpton 322 cuneiform tablet, some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, the Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. Ahmes solution to the problem is the ratio of half the side of the base of the pyramid to its height, in other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face. Ancient Greek and Hellenistic mathematicians made use of the chord, given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector passes through the center of the circle and bisects the angle. One half of the chord is the sine of one half the bisected angle, that is, c h o r d θ =2 sin θ2. Due to this relationship, a number of identities and theorems that are known today were also known to Hellenistic mathematicians. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively, theorems on the lengths of chords are applications of the law of sines. And Archimedes theorem on broken chords is equivalent to formulas for sines of sums, the first trigonometric table was apparently compiled by Hipparchus of Nicaea, who is now consequently known as the father of trigonometry. Hipparchus was the first to tabulate the corresponding values of arc and it seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords
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Uses of trigonometry
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The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics. In fine, it is the soul of science and it is an eternal truth, it contains the mathematical demonstration of which man speaks, and the extent of its uses are unknown. For the 25 years preceding the invention of the logarithm in 1614 and it used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those angles. It does mean that things in these fields cannot be understood without trigonometry. For example, a professor of music may perhaps know nothing of mathematics, in some of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. The resemblance between the shape of a string and the graph of the sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some waves and the graph of the function is also not coincidental. In some other fields, among them climatology, biology, and economics, the study of these often involves the periodic nature of the sine and cosine function. Many fields make use of trigonometry in more advanced ways than can be discussed in a single article, often those involve what are called Fourier series, after the 18th- and 19th-century French mathematician and physicist Joseph Fourier. Fourier used these for studying heat flow and diffusion, Fourier series are also applicable to subjects whose connection with wave motion is far from obvious. Another example, mentioned above, is diffusion, among others are, the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenbergs inequality. A more abstract concept than Fourier series is the idea of Fourier transform, Fourier transforms involve integrals rather than sums, and are used in a similarly diverse array of scientific fields. Many natural laws are expressed by relating rates of change of quantities to the quantities themselves, for example, The rate of change of population is sometimes jointly proportional to the present population and the amount by which the present population falls short of the carrying capacity. This kind of relationship is called a differential equation, if, given this information, one tries to express population as a function of time, one is trying to solve the differential equation. Fourier transforms may be used to convert some differential equations to algebraic equations for which methods of solving them are known, in almost any scientific context in which the words spectrum, harmonic, or resonance are encountered, Fourier transforms or Fourier series are nearby. Intelligence quotients are sometimes held to be distributed according to the bell-shaped curve, about 40% of the area under the curve is in the interval from 100 to 120, correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under the curve is in the interval from 120 to 140, correspondingly, similarly many other things are distributed according to the bell-shaped curve, including measurement errors in many physical measurements
18.
Trigonometric functions
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
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Inverse trigonometric functions
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In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations used for the trigonometric functions. The most common convention is to name inverse trigonometric functions using a prefix, e. g. arcsin, arccos, arctan. This convention is used throughout the article, when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Similarly, in programming languages the inverse trigonometric functions are usually called asin, acos. The notations sin−1, cos−1, tan−1, etc, the confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, −1 = sec. Nevertheless, certain authors advise against using it for its ambiguity, since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. There are multiple numbers y such that sin = x, for example, sin =0, when only one value is desired, the function may be restricted to its principal branch. With this restriction, for x in the domain the expression arcsin will evaluate only to a single value. These properties apply to all the trigonometric functions. The principal inverses are listed in the following table, if x is allowed to be a complex number, then the range of y applies only to its real part. Trigonometric functions of trigonometric functions are tabulated below. This is derived from the tangent addition formula tan = tan + tan 1 − tan tan , like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative,11 − z 2, as a binomial series, the series for arctangent can similarly be derived by expanding its derivative 11 + z 2 in a geometric series and applying the integral definition above. Arcsin = z + z 33 + z 55 + z 77 + ⋯ = ∑ n =0 ∞, for example, arccos x = π /2 − arcsin x, arccsc x = arcsin , and so on. Alternatively, this can be expressed, arctan z = ∑ n =0 ∞22 n 2. There are two cuts, from −i to the point at infinity, going down the imaginary axis and it works best for real numbers running from −1 to 1
20.
Trigonometric constants expressed in real radicals
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Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. When they are, they are more specifically in terms of square roots. For an angle of a number of degrees, which is not a multiple 3°, the values of sine, cosine. Note that 1° = π/180 radians, according to Nivens theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2,1, −1/2, and −1. According to Bakers theorem, if the value of a sine and that is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values. The list in this article is incomplete in several senses, first, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here. Second, it is possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle. This article only gives the cases based on the Fermat primes 3 and 5, thus for example cos, given in the article 17-gon, is not given here. Fourth, this article deals with trigonometric function values when the expression in radicals is in real radicals—roots of real numbers. Many other trigonometric function values are expressible in, for example, in practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Generating trigonometric tables. Several different units of measure are widely used, including degrees, radians. The following table shows the conversions and values for some common angles, Values outside the range are trivially derived from these values. This is because the sum of the angles of any n-gon is 180° ×, using cos 36 ∘ =5 +14, tan 36 ∘ =5 −25, this can be simplified to, V = a 34. The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility of right triangles, here right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a polygon, a vertex, an edge center containing that vertex. 2 sin θ =2 −2 cos 2 θ =2 −2 +2 cos 4 θ =2 −2 +2 +2 cos 8 θ and so on. If M =2 and N =2 then cos π17 = M −4 +28, crd is the chord function, crd θ =2 sin θ2. Thus sin 18 ∘ =11 +5 =5 −14, similarly crd 108 ∘ = crd = b a =1 +52, so sin 54 ∘ = cos 36 ∘ =1 +54
21.
Trigonometric tables
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In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science, the calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices. Modern computers and pocket calculators now generate trigonometric function values on demand, often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of functions is still used in computer graphics. In this case, calling generic library routines every time is unacceptably slow, one option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT, modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC that is efficient, since it uses only shifts. All of these methods are implemented in hardware for performance reasons. The particular polynomial used to approximate a trig function is generated ahead of time using some approximation of an approximation algorithm. Trigonometric functions of angles that are multiples of 2π are algebraic numbers. The values for a/b·2π can be found by applying de Moivres identity for n = a to a bth root of unity, for this case, a root-finding algorithm such as Newtons method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however and this method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy. In modern form, the identities he derived are stated as follows, unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/N. For example, for N =256 the maximum error in the values is ~0.061. For N =1024, the error in the sine values is ~0.015. If the sine and cosine values obtained were to be plotted, N −1, where wr = cos and wi = sin. These two starting trigonometric values are computed using existing library functions
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Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
23.
Law of sines
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In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. When the last of these equations is not used, the law is sometimes stated using the reciprocals, the law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. Numerical calculation using this technique may result in an error if an angle is close to 90 degrees. It can also be used when two sides and one of the angles are known. In some such cases, the triangle is not uniquely determined by this data, the law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature, the area T of any triangle can be written as one half of its base times its height. Thus, depending on the selection of the base the area of the triangle can be written as any of, multiplying these by 2/abc gives 2 T a b c = sin A a = sin B b = sin C c. When using the law of sines to find a side of a triangle, in the case shown below they are triangles ABC and AB′C′. Given a general triangle the following conditions would need to be fulfilled for the case to be ambiguous, The only information known about the triangle is the angle A, the side a is shorter than the side c. The side a is longer than the altitude h from angle B, without further information it is impossible to decide which is the triangle being asked for. The following are examples of how to solve a problem using the law of sines, given, side a =20, side c =24, and angle C = 40°. Using the law of sines, we conclude that sin A20 = sin 40 ∘24, note that the potential solution A =147. 61° is excluded because that would necessarily give A + B + C > 180°. The second equality above readily simplifies to Herons formula for the area, the law of sines takes on a similar form in the presence of curvature. In the spherical case, the formula is, sin A sin α = sin B sin β = sin C sin γ. Here, α, β, and γ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a, b, and c, respectively, a, B, and C are the surface angles opposite their respective arcs. See also Spherical law of cosines and Half-side formula, in hyperbolic geometry when the curvature is −1, the law of sines becomes sin A sinh a = sin B sinh b = sin C sinh c. Define a generalized function, depending also on a real parameter K. The law of sines in constant curvature K reads as sin A sin K a = sin B sin K b = sin C sin K c
24.
Law of cosines
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In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi, the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and it is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and C =. By the distance formula, we have c =2 +2, an advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
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Law of tangents
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In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that a − b a + b = tan tan , to prove the law of tangents we can start with the law of sines, a sin α = b sin β. Let d = a sin α, d = b sin β so that a = d sin α and b = d sin β. As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity tan = sin α ± sin β cos α + cos β. The law of tangents can be used to compute the missing side, on a sphere of unit radius, the sides of the triangle are arcs of great circles. Accordingly their lengths can be expressed in radians or any other units of angular measure, let A, B, C be the angles at the three vertices of the triangle and let a, b, c be the respective lengths of the opposite sides. The spherical law of tangents says tan tan = tan tan , Law of sines Law of cosines Law of cotangents Mollweides formula Half-side formula Tangent half-angle formula
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Law of cotangents
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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. In the upper figure, the points of tangency of the incircle with the sides of the break the perimeter into 6 segments. In each pair the segments are of equal length, for example, the 2 segments adjacent to vertex A are equal. If we pick one segment from each pair, their sum will be the semiperimeter s, an example of this is the segments shown in color in the figure. The two segments making up the red line add up to a, so the blue segment must be of length s − a. Obviously, the five segments must also have lengths s − a, s − b, or s − c. By inspection of the figure, using the definition of the cotangent function, we have cot = s − a r, a number of other results can be derived from the law of cotangents. Note that the area of triangle ABC is also divided into 6 smaller triangles, also in 3 pairs, for example, the two triangles near vertex A, being right triangles of width s − a and height r, each have an area of 1/2r. From the addition formula and the law of cotangents we have sin sin = cot − cot cot + cot = a − b 2 s − a − b. This gives the result a − b c = sin cos as required, here, an extra step is required to transform a product into a sum, according to the sum/product formula. This gives the result b + a c = cos sin as required, the law of tangents can also be derived from this. Law of sines Law of cosines Law of tangents Mollweides formula Formula sheet database – law of cotangents, silvester, John R. Geometry, Ancient and Modern
27.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
28.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
29.
Differentiation of trigonometric functions
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The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin, cos and tan, for example, the derivative of f = sin is represented as f ′ = cos. F ′ is the rate of change of sin at a point a. All derivatives of trigonometric functions can be found using those of sin. The quotient rule is implemented to differentiate the resulting expression. Finding the derivatives of the trigonometric functions involves using implicit differentiation. Let θ be the angle at O made by the two radii OA and OB, since we are considering the limit as θ tends to zero, we may assume that θ is a very small positive number,0 < θ ≪1. Consider the following three regions of the diagram, R1 is the triangle OAB, R2 is the circular sector OAB, clearly, Area < Area < Area. Using basic trigonometric formulae, the area of the triangle OAB is 12 × | | O A | | × | | O B | | × sin θ =12 r 2 sin θ. Collecting together these three areas gives, Area < Area < Area ⟺12 r 2 sin θ <12 r 2 θ <12 r 2 tan θ, since r >0, we can divide through by ½·r2. This means that the construction and calculations are all independent of the circles radius, in the last step we simply took the reciprocal of each of the three terms. Since all three terms are positive this has the effect of reversing the inequities, e. g. if 2 <3 then ½ > ⅓. We have seen that if 0 < sin θ ≪1 then sin/θ is always less than 1 and, notice that as θ gets closer to 0, so cos θ gets closer to 1. Informally, as θ gets smaller, sin/θ is squeezed between 1 and cos θ, which itself it heading towards 1 and it follows that sin/θ tends to 1 as θ tends to 0 from the positive side. The last section enables us to calculate this new limit relatively easily and this is done by employing a simple trick. In this calculation, the sign of θ is unimportant, lim θ →0 = lim θ →0 = lim θ →0. The well-known identity sin2θ + cos2θ =1 tells us that cos2θ –1 = –sin2θ, to calculate the derivative of the sine function sin θ, we use first principles. By definition, d d θ sin θ = lim δ →0, using the well-known angle formula sin = sin α cos β + sin β cos α, we have, d d θ sin θ = lim δ →0 = lim δ →0
30.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
31.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
32.
Right triangle
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A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a triangle is the basis for trigonometry. The side opposite the angle is called the hypotenuse. The sides adjacent to the angle are called legs. Side a may be identified as the adjacent to angle B and opposed to angle A, while side b is the side adjacent to angle A. If the lengths of all three sides of a triangle are integers, the triangle is said to be a Pythagorean triangle. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the then the other is height. As a formula the area T is T =12 a b where a and b are the legs of the triangle and this formula only applies to right triangles. From this, The altitude to the hypotenuse is the mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse, in equations, f 2 = d e, b 2 = c e, a 2 = c d where a, b, c, d, e, f are as shown in the diagram. Moreover, the altitude to the hypotenuse is related to the legs of the triangle by 1 a 2 +1 b 2 =1 f 2. For solutions of this equation in integer values of a, b, f, the altitude from either leg coincides with the other leg. Since these intersect at the vertex, the right triangles orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that, In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs. This can be stated in equation form as a 2 + b 2 = c 2 where c is the length of the hypotenuse, Pythagorean triples are integer values of a, b, c satisfying this equation. The radius of the incircle of a triangle with legs a and b. The radius of the circumcircle is half the length of the hypotenuse, thus the sum of the circumradius and the inradius is half the sum of the legs, R + r = a + b 2
33.
Hypotenuse
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In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. For example, if one of the sides has a length of 3. The length of the hypotenuse is the root of 25. The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus 54d, a folk etymology says that tenuse means side, so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle are a and b and that of the hypotenuse is c, many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow, some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the line at the same time when given x and y. The angle returned will normally be given by atan2. Orthographic projections, The length of the hypotenuse equals the sum of the lengths of the projections of both catheti. And The square of the length of a cathetus equals the product of the lengths of its projection on the hypotenuse times the length of this. Given the length of the c and of a cathetus b. The adjacent angle of the b, will be α = 90° – β One may also obtain the value of the angle β by the equation. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. Hypotenuse
34.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
35.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
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Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
37.
Periodic function
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In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P
38.
Sound
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In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a transmission medium such as air or water. In physiology and psychology, sound is the reception of such waves, humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and below 20 Hz is infrasound, other animals have different hearing ranges. Acoustics is the science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound. A scientist who works in the field of acoustics is an acoustician, an audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound. Auditory sensation evoked by the oscillation described in, sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium, as the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, at an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave and this is intuitively obvious for a solid, and the same is true for liquids and gases. During propagation, waves can be reflected, refracted, or attenuated by the medium, the behavior of sound propagation is generally affected by three things, A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium, if the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the wave will be decreased by the speed of the wind. Medium viscosity determines the rate at which sound is attenuated, for many media, such as air or water, attenuation due to viscosity is negligible. When sound is moving through a medium that does not have constant physical properties, the mechanical vibrations that can be interpreted as sound can travel through all forms of matter, gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium, sound cannot travel through a vacuum. Sound is transmitted through gases, plasma, and liquids as longitudinal waves and it requires a medium to propagate
39.
Gupta period
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The Gupta Empire was an ancient Indian empire founded by Sri Gupta. The empire existed at its zenith from approximately 320 to 550 CE, the peace and prosperity created under the leadership of the Guptas enabled the pursuit of scientific and artistic endeavors. Chandragupta I, Samudragupta, and Chandragupta II were the most notable rulers of the Gupta dynasty, the high points of the period is great cultural developments which took place during the reign of Chandragupta II. Science and political administration reached new heights during the Gupta era, strong trade ties also made the region an important cultural center and set the region up as a base that would influence nearby kingdoms and regions in Burma, Sri Lanka, and Southeast Asia. The earliest available Indian epics are also thought to have committed to written texts around this period. After the collapse of the Gupta Empire in the 6th century, a minor line of the Gupta clan continued to rule Magadha after the disintegration of the empire. These Guptas were ultimately ousted by Vardhana ruler Harsha, who established his empire in the first half of the 7th century, according to many historians, the Gupta dynasty was a Vaishya dynasty. Historian Ram Sharan Sharma asserts that the Vaishya Guptas appeared as a reaction against oppressive rulers, brannigan, the rise of the Gupta Empire was one of the most prominent violations of the caste system in ancient India. There is controversy among scholars about the homeland of the Guptas. Jayaswal has pointed out that the Guptas were originally inhabitants of Prayaga, Uttar Pradesh, in north India, another scholar, Gayal supported the theory of Jaiswal, suggesting that the original home of the Guptas was Antarvedi embracing the regions of Oudh and Prayag. However another historian of this time in Indian history, Ganguli, has offered a different view about the original Gupta homeland, according to him the Guptas homeland is further south, the Murshidabad region of Bengal, and not Magadha in Bihar. He based his theory on the statement of the Chinese Buddhist monk, Yijing, fleet and other historians however criticize Gangulis theory because Sri Gupta ruled during the end of the 3rd century, but Yijing placed him at the end of the 2nd century. Hence the theory of historians, who have provided their views based on the accounts of Yijing, are considered less valid than theories based on sources such as coinage. From these theories, several conflicting opinions about the original homeland, according to Allan and a few other scholars, the Guptas were initially concentrated in the region of Magadha and from there they extended their sway to Bengal. According to other groups, the homeland of the Guptas was Varendri or the Varendra Bhumi in Bengal. Whatever the theory is, the rule of the Guptas initiated the Golden Age in history of ancient India, bengali historians like HC Raychoudhuri the Guptas originated from the Varendri region which is now part of Rangpur and Rajshahi Division of modern-day Bangladesh. DC Ganguly, on the hand, considers the surrounding region of Murshidabad as the original home of the Guptas. The most likely time for the reign of Sri Gupta is c, the Murundas who were feudal lords of Kushans provided or granted land to Srigupta
40.
Indian astronomy
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Indian astronomy has a long history stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley Civilization or earlier, astronomy later developed as a discipline of Vedanga or one of the auxiliary disciplines associated with the study of the Vedas, dating 1500 BCE or older. The oldest known text is the Vedanga Jyotisha, dated to 1400–1200 BCE, as with other traditions, the original application of astronomy was thus religious. Indian astronomy flowered in the 5th-6th century, with Aryabhata, whose Aryabhatiya represented the pinnacle of astronomical knowledge at the time, Later the Indian astronomy significantly influenced Muslim astronomy, Chinese astronomy, European astronomy, and others. Other astronomers of the era who further elaborated on Aryabhatas work include Brahmagupta, Varahamihira. Some of the earliest forms of astronomy can be dated to the period of Indus Valley Civilization or earlier, some cosmological concepts are present in the Vedas, as are notions of the movement of heavenly bodies and the course of the year. Thus, the Shulba Sutras, texts dedicated to altar construction, discusses advanced mathematics, Vedanga Jyotisha is another of the earliest known Indian texts on astronomy, it includes the details about the sun, moon, nakshatras, lunisolar calendar. Greek astronomical ideas began to enter India in the 4th century BCE following the conquests of Alexander the Great, by the early centuries of the Common Era, Indo-Greek influence on the astronomical tradition is visible, with texts such as the Yavanajataka and Romaka Siddhanta. Later astronomers mention the existence of various siddhantas during this period and these were not fixed texts but rather an oral tradition of knowledge, and their content is not extant. The text today known as Surya Siddhanta dates to the Gupta period and was received by Aryabhata, the classical era of Indian astronomy begins in the late Gupta era, in the 5th to 6th centuries. The Pañcasiddhāntikā by Varāhamihira approximates the method for determination of the direction from any three positions of the shadow using a gnomon. By the time of Aryabhata the motion of planets was treated to be rather than circular. The divisions of the year were on the basis of religious rites, in the Vedānga Jyotiṣa, the year begins with the winter solstice. Hindu calendars have several eras, The Hindu calendar, counting from the start of the Kali Yuga, has its epoch on 18 February 3102 BCE Julian, the Vikrama Samvat calendar, introduced about the 12th century, counts from 56–57 BCE. The Saka Era, used in some Hindu calendars and in the Indian national calendar, has its epoch near the equinox of year 78. The Saptarshi calendar traditionally has its epoch at 3076 BCE and this device finds mention in the works of Varāhamihira, Āryabhata, Bhāskara, Brahmagupta, among others. The Cross-staff, known as Yasti-yantra, was used by the time of Bhaskara II and this device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. The clepsydra was used in India for astronomical purposes until recent times, Ōhashi notes that, Several astronomers also described water-driven instruments such as the model of fighting sheep
41.
Surya Siddhanta
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The Surya Siddhanta is the name of multiple treatises in Indian astronomy. The extant text as translated by Burgess is medieval, but it is based on older versions. It has rules laid down to determine the true motions of the luminaries and it gives the locations of several stars other than the lunar nakshatras and treats the calculation of solar eclipses as well as solstices, e. g. summer solstice 21/06. Significant coverage is on kinds of time, length of the year of devas and asuras, day and night of Brahma, the Earths diameter and circumference are also given. Eclipses and color of the portion of the moon are mentioned. Judging from the dates in the work, Plofker suggests that this Sūrya-siddhānta was composed or revised in the early sixth century. Utpala, a 10th-century commentator of Varahamihira, quotes six shlokas of the Surya Siddhanta of his day, the present version was modified by Bhaskaracharya during the Middle Ages. It is partly based on Vedanga Jyotisha, which itself might reflect traditions going back to the Indian Iron Age and it is hypothesized that there were cultural contacts between the Indian and Greek astronomers via cultural contact with Hellenistic Greece, specifically the work of Hipparchus. There were many similarities between Suryasiddhanta and Greek astronomy in Hellenistic period, for example, Suryasiddhanta provides more accurate and detailed table of sines than Hipparchus. However, the model of Suryasiddhanta was simpler than that made by Ptolemy in the 2nd century. The astronomical time cycles contained in the text were remarkably accurate at the time, the Hindu Time Cycles, copied from an earlier work, are described in verses 11–23 of Chapter 1,11. That which begins with respirations is called real, six respirations make a vinadi, sixty of these a nadi,12. And sixty nadis make a day and night. Of thirty of these days is composed a month, a civil month consists of as many sunrises,13. A lunar month, of as many days, a solar month is determined by the entrance of the sun into a sign of the zodiac. This is called a day of the gods, the day and night of the gods and of the demons are mutually opposed to one another. Six times sixty of them are a year of the gods, twelve thousand of these divine years are denominated a caturyuga, of ten thousand times four hundred and thirty-two solar years 16. Is composed that caturyuga, with its dawn and twilight, the difference of the krtayuga and the other yugas, as measured by the difference in the number of the feet of Virtue in each, is as follows,17
42.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
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Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
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Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
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Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings