Trigonometrical function

Definitions

  • Webster's Revised Unabridged Dictionary
    • Trigonometrical function a quantity whose relation to the variable is the same as that of a certain straight line drawn in a circle whose radius is unity, to the length of a corresponding are of the circle. Let AB be an arc in a circle, whose radius OA is unity let AC be a quadrant, and let OC, DB, and AF be drawnpependicular to OA, and EB and CG parallel to OA, and let OB be produced to G and F. E Then BD is the sine of the arc AB; OD or EB is the cosine, AF is the tangent, CG is the cotangent, OF is the secant OG is the cosecant, AD is the versed sine, and CE is the coversed sine of the are AB. If the length of AB be represented by xOA being unity) then the lengths of Functions. these lines (OA being unity) are the trigonometrical functions of x, and are written sin x cos x tan xor tang x), cot x sec x cosec x versin x coversin x. These quantities are also considered as functions of the angle BOA.
    • Trigonometrical function See under Function.
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Usage

In literature:

Roger Cotes (1722) was the first to differentiate a trigonometrical function.
"Encyclopaedia Britannica, 11th Edition, Volume 14, Slice 5" by Various
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In science:

In fact, it is simple to construct a CS scalar, such as a trigonometric function of spacetime, and show that for such a scalar the C-tensor does not vanish, thus rendering the Schwarzschild metric not a solution of CS modified gravity, in spite of the vanishing of the Pontryagin density.
Chern-Simons Modified General Relativity
In certain limits appropriate to inflation, one can solve this differential equation in terms of Whittaker functions, which can be decomposed into products of trigonometric functions and exponentials.
Chern-Simons Modified General Relativity
Furthermore, as was discussed with the example of trigonometric functions, there is no such thing as a definition not being exact.
Causal set as a discretized phase spacetime
Following [8, §8.2], R ∞ 0 eU ′ (t)H (t)e−qt dt = R ∞ and the fact that eU (t) is a continuously differentiable function for all t ∈ R (e.g., a polynomial or trigonometric function in Stokes-type problems), we have the desired result.
Comments on: "Energetic balance for the Rayleigh--Stokes problem of an Oldroyd-B fluid" [Nonlinear Anal. RWA 12 (2011) 1]
In IX is given by the integral of (p, q)-trigonometric functions.
On the structural stability of planar quasihomogeneous polynomial vector fields
For (p, q) = (1, 1), we have that Csφ = cos φ, Snφ = sin φ, i.e. trigonometric functions are the classical ones.
On the structural stability of planar quasihomogeneous polynomial vector fields
In view of this equation and of Eq. (32) for cos ∆k one can conveniently express all the trigonometric functions of δk and θk in Eqs. (A6) and (A10) in terms of cos(2δk ) and cos(2θk ).
Dynamic correlations, fluctuation-dissipation relations, and effective temperatures after a quantum quench of the transverse field Ising chain
This classification states that all such solutions are unitary and, in terms of dependence on u, there are only three types of functions r(u): rational, trigonometric, and elliptic.
Representations of affine Lie algebras, elliptic r-matrix systems, and special functions
Then we define twisted correlation functions and show that they satisfy a twisted trigonometric KZ system.
Representations of affine Lie algebras, elliptic r-matrix systems, and special functions
The function Ψ satisfies a twisted version of the trigonometric KZ equations.
Representations of affine Lie algebras, elliptic r-matrix systems, and special functions
Its matrix elements are trigonometric rational functions of (cid:21). .
Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials
Its matrix elements are trigonometric rational functions of (cid:21). .
Representation-theoretic proof of the inner product and symmetry identities for MacDonald's polynomials
In the argument of the trigonometric functions, the operator ←−∇ r acts on the electron plasma frequency, and the operator −→∇ k acts on the components of the Wigner matrix W (k, r, t).
Wigner-Moyal description of free variable mass Klein-Gordon fields
Another trend is based on the expansion of functions on suitable bases, formerly the trigonometric basis, more recently bases derived from a multiresolution analysis (wavelet bases and the like).
The Brouwer Lecture 2005: Statistical estimation with model selection
Every 2π -periodic continuous matrix function can be uniformly approximated by trigonometric matrix polynomials.
Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle
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