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A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: [4] they admit nontrivial factorizations in Z[ω]. For example: 3 = −(1 + 2ω) 2 7 = (3 + ω)(2 − ω). In general, if a natural prime p is 1 modulo 3 and can therefore be written as p = a 2 − ab + b 2, then it factorizes over Z[ω] as p = (a + bω)((a − b ...
Bézout's identity (despite its usual name, it is not, properly speaking, an identity) Binet-cauchy identity; Binomial inverse theorem; Binomial identity; Brahmagupta–Fibonacci two-square identity; Candido's identity; Cassini and Catalan identities; Degen's eight-square identity; Difference of two squares; Euler's four-square identity; Euler ...
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [1] [2]A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel [] in 1746.
The function f(x) = x β (with β ≤ 1) defined on [0, 1] serves as a prototypical example of a function that is C 0,α Hölder continuous for 0 < α ≤ β, but not for α > β. Further, if we defined f analogously on [,), it would be C 0,α Hölder continuous only for α = β. If a function is –Hölder continuous on an interval and >, then ...
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The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function (;,) of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n: