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Exact Trigonometric Constants
In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers. Common angles The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45° (see below for proofs). In the table below, the label "Undefined" represents a ratio 1:0. If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their multiplicative inverse, reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding Inverse trigonometric functions, inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roots Of Unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation z^n = 1. Unless otherwise specified, the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in More specifically, ''J''''α'' is the kernel of the ring homomorphism from ''F'' 'x''to ''E'' which sends polynomials ''g'' to their value ''g''(''α'') at the element ''α''. Because it is the kernel of a ring homom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Trigonometric Identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is :\sin^2 \theta + \cos^2 \theta = 1. As usual, \sin^2 \theta means (\sin\theta)^2. Proofs and their relationships to the Pythagorean theorem Proof based on right-angle triangles Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely . The elementary definitions of the sine and cosine functions in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Prime
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ... . If 2''k'' + 1 is prime and , then ''k'' itself must be a power of 2, so is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are , , , , and . Basic properties The Fermat numbers satisfy the following recurrence relations: : F_ = (F_-1)^+1 : F_ = F_ \cdots F_ + 2 for ''n'' ≥ 1, : F_ = F_ + 2^F_ \cdots F_ : F_ = F_^2 - 2(F_-1)^2 for . Each of these relations can be proved by mathematical induction. From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that and ''F'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hierarchy, is exactly equal to H_(1). Powers of two with Sign (mathematics)#Terminology for signs, non-negative exponents are integers: , , and is two multiplication, multiplied by itself times. The first ten powers of 2 for non-negative values of are: :1, 2, 4, 8, 16 (number), 16, 32 (number), 32, 64 (number), 64, 128 (number), 128, 256 (number), 256, 512 (number), 512, ... By comparison, powers of two with negative exponents are fractions: for positive integer , is one half multiplied by itself times. Thus the first few negative powers of 2 are , , , , etc. Sometimes these are called ''inverse powers of two'' because each is the multiplicative inverse of a positive power of two. Base of the binary numeral sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Prime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-Year College Mathematics Journal
The ''College Mathematics Journal'' is an expository magazine aimed at teachers of college mathematics, particularly those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is a continuation of the ''Two-Year College Mathematics Journal''. It covers all aspects of mathematics. It publishes articles intended to enhance undergraduate instruction and classroom learning, including expository articles, short notes, problems, and "mathematical ephemera" such as fallacious proofs, quotations, cartoons, poetry, and humour. Paid circulation in 2008 was 9,000, and total circulation was 9,500. The MAA gives the George Pólya Awards annually "for articles of expository excellence" published in the ''College Mathematics Journal''. References External links *''The College Mathematics Journal''at JSTOR''The College Mathematics Journal''at Taylor & Francis Taylor & Francis Group is an international company origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niven's Theorem
In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of in the interval for which the sine of ' degrees is also a rational number are: : \begin \sin 0^\circ & = 0, \\0pt\sin 30^\circ & = \frac 12, \\0pt\sin 90^\circ & = 1. \end In radians, one would require that , that be rational, and that be rational. The conclusion is then that the only such values are , , and . The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. The theorem extends to the other trigonometric functions as well. For rational values of , the only rational values of the sine or cosine are , , and ; the only rational values of the secant or cosecant are and ; and the only rational values of the tangent or cotangent are and .A proof for the cosine case appears as Lemma 12 in History Niven's proof of his theorem appears in his book ''Irrational Numbers''. Earlier, the theorem had been proven by D. H. Lehmer and J. M. H. Olmstead. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindemann–Weierstrass Theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transcendence degree over \mathbb. An equivalent formulation from , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over \mathbb by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial Of 2cos(2pi/n)
In number theory, the real parts of the roots of unity are related to one-another by means of the minimal polynomial of 2\cos(2\pi/n). The roots of the minimal polynomial are twice the real part of the roots of unity, where the real part of a root of unity is just \cos\left(2k\pi/n\right) with k coprime with n. Formal definition For an integer n \geq 1, the minimal polynomial \Psi_n(x) of 2\cos(2\pi/n) is the non-zero integer-coefficient monic polynomial of smallest degree for which \Psi_n\!\left(2\cos(2\pi/n)\right) = 0. For every , the polynomial \Psi_n(x) is monic, has integer coefficients, and is irreducible over the integers and the rational numbers. All its roots are real; they are the real numbers 2\cos\left(2k\pi/n\right) with k coprime with n and either 1 \le k 1 of n, including n itself: :\prod _\Psi_( x)=\chi_( x). This means that the \Psi_(x) are exactly the irreducible factors of \chi_(x), which allows to easily obtain \Psi_(x) for any odd d, knowing its degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
