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This article deals with trigonometric functional equations on hypergroups. We describe the general continuous solution of sine and cosine addition formulas and a so-called sine-cosine functional equation on a locally compact hypergroup in terms of exponential functions, sine functions, and second-order generalized moment functions.

We present the (co)sine n-algebra which is indexed by the d-dimensional integer lattice. Due to the associative operators, this generalized (co)sine n-algebra is the higher order Lie algebra for the n even case. The particular cases are the d-dimensional lattice sine 3 and cosine 5-algebras with the special parameter values. We find that the corresponding d-dimensional lattice sine 3 and cosine 5-algebras are the Nambu 3-algebra and higher order Lie algebra, respectively. The limiting case of the d-dimensional lattice (co)sine n-algebra is also discussed. Moreover we construct the super sine n-algebra, which is the super higher order Lie algebra for the n even case.

We consider a class of integral equations characterized by kernels of cosine and sine type and study their solvability. Moreover, we analyse the integral operator T, which is generating those equations, by identifying its characteristic polynomial, characterizing its invertibility, spectrum, Parseval type identity and involution properties. Additionally, a new convolution is here introduced, associated with T, for which we deduce a corresponding factorization property.

The student who has learned the binomial expansion and the formula for exp (ix), is shown how to obtain simple expansions for sin nx, cos nx, sinh nx, cosh nx, sin αx and cos αx for n any positive integer and α any real number.

We introduce a new kind of double sequences named LBVDS N and generalize Moricz’s theorem.

We construct the exact finite difference equation discretizations for the nonlinear differential equations whose solutions are the Jacobi cosine and sine functions. Our derivations clarify and extend previous work done on this topic.

In this paper, we introduce fractional cosine and sine transforms over finite fields. The basis for the definition of such transforms, which are respectively referred by the acronyms GFrCT and GFrST, is the recently proposed fractional Fourier transform over finite fields (GFrFT) [1] . More specifically, we use the eigenvectors of the finite field Fourier transform to construct eigenvectors of the finite field cosine and sine transforms. Such eigenvectors are then used in spectral expansions that allow to compute fractional powers of the finite field cosine and sine transform matrices.

In this paper we introduce some new modified cosine sums and then using these sums we study $L^{1}$-convergence of trigonometric cosine series.

In this paper, have study new results of half canonical cosine and sine transforms of generalized function. Keyword: Canonical transform, half canonical cosine transform, half canonical sine transform, testing function space.

We show that high-dimensional analogues of the sine function (more precisely, the d-dimensional polar sine and the d-th root of the d-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H. Adopting the language of Deza and Rosenberg, we say that these d-dimensional sine functions are d-semimetrics. We also establish geometric identities for both the d-dimensional polar sine and the d-dimensional hypersine. We then show that when d=1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the d-dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms ''with high probability''.

Our aim is to investigate the behavior of generalized multiple sine functions with general period parameters in the fundamental domain. For that, we need to calculate the number of their extremal values. By estimating their special values, we determine it in some cases including the quintuple sine function. As a consequence, we sketch their graphs.

We investigate values of the multiple sine function of period (1,...,1) at rational numbers. We show explicit formulas using special values of Dirichlet L-functions. We also study the case of the multiple gamma function.

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.

In this paper, we introduce a Daehee constant, the so-called q-extension of the Napier constant, and consider the Daehee formula associated with the q-extensions of trigonometric functions. That is, we derive the q-extensions of sine and cosine functions from our Daehee formula. Finally, we present the q-calculus related to the q-extensions of sine and cosine functions.

Here we present Ostrowski type inequalities on Cosine and Sine Operator Functions for various norms.

In this paper the sine and cosine functional equations are considered and solved on arbitrary polynomial hypergroups in a single variable. The method of solution is based on spectral synthesis.

We give a short proof to characterize the cases when arccos(r), the arccosine of the squareroot of a rational number r [0, 1], is a rational multiple of : This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind. © Versita Warsaw and Springer-Verlag Berlin Heidelberg 2006.

For functions of two variables defined by trigonometric cosine series with quasiconvex coefficients, we obtain estimates for their variations in the Hardy–Vitali sense.

Hyperbolic sine and cosine functions on measure chains are introduced and their basic properties are established. A necessary and sufficient condition for (non)existence of focal points (generalized zeroes) of the hyperbolic cosine and of particular exponential functions on measure chains is found.

We introduce generalized convolutions of Stieltjes, Hilbert, and Fourier sine and cosine transforms and consider their applications to integral equations.

Some results concerning generalized Fourier sine- and cosine- transforms are discussed. The well known sine (or cosine) Fourier transform is an isometric mapping of L2 (0,∞) on itself (see [1]). It is interesting to consider the expression of a function with respect to sin(αξ + ϕ), where ϕ is a constant. Such an approach can be found in the papers of G.H. Hardy [2], R.G. Cooke [3], in [4] (the formula 7.10), and in a recent paper by A. Zilberglat and N. Lebedev [5]. In these works it was shown that an integrable function on (0, ∞) of a bounded variation over (0, ∞) can be repersented in the form of an integral of a hypergeometric function. In th Paper we consider generalized Fourier sine- and cosine- transforms of functions belonging to the space L2 (0, ∞). We shown the uniqueness and continuity of such an representation. Moreover, we obtain relations between formulae from the papers [2], [3] and [5].

We propose a new way of calculating the sine and cosine functions. The method is based on recursive applications of a modified complex power algorithm. On a machine with multiple complex multipliers the method can be used to calculate sines and cosines in logarithmic time. The serial version of the presented method requires only two precomputed constants and no tables. In the parallel versions a trade off can be made between the number of parallel processing elements and the size of tables.

Trigonometric matrix functions play a fundamental role in second order differential equation systems. This work presents an algorithm for computing the cosine matrix function based on Taylor series and the cosine double angle formula. It uses a forward absolute error analysis providing sharper bounds than existing methods. The proposed algorithm had lower cost than state-of-the-art algorithms based on Hermite matrix polynomial series and Padé approximants with higher accuracy in the majority of test matrices., This work has been supported by Universitat Politecnica de Valencia Grant PAID-06-011-2020.

Considering the characteristic of the angles, the sine and cosine values of which compose the discrete- sine and cosine wave, the new improved algorithm is presented. In the algorithm, the theory of setting Renminbi notes is used to selecting the basic angle sequence and the symmetry of trigonometric function is used to compute the sine and cosine values of the basic angle expressed as 2k?. Simulation results prove that the precision may reach 10^-6 under 5 iterations. Compared with other algorithms, The computational complexity of the algorithm is simple, the precise is higher. Therefore the algorithm is feasible for the NCO.

Crash pulses in automotive collisions often exhibit acceleration shapes somewhere between a sine and a step function and velocity shapes somewhere between a cosine and a linear decay. This is an example of real world behavior that is only somewhat like the familiar sine, cosine, or tangent shapes so commonly used in physical modeling. To adjust the mathematics to the problem, two familiar ordinary differential equations are merged to create a mathematical transition between trigonometric functions and polynomials by introducing one new parameter. The merged ODE produces a new set of “transitional trigonometry” functions that include both sets of familiar shapes and everything in between. For example, the sine function transitions smoothly into a constant or step function. The corresponding cosine function becomes a straight line. When the sine and cosine are plotted against each other the familiar unit circle undergoes a metamorphosis into a square. Integrals of these transitional trigonometric functions transition into a parabola, cubic polynomial, etc. These functions were developed to model a crash pulse in a vehicle collision, a task for which they work remarkably well. Basically, these functions are able to model a structure with force-deflection properties somewhere between a spring with linearly increasing force and a device that produces a constant force. One wonders what other applications in physics may exist besides crashing cars and what other pairs of physical models (represented by ODEs) might be merged together to produce other new and useful transitions.Copyright © 2008 by ASME

In this paper, we give connections between distribution cosine functions (defined in [10]) and almost-distribution cosine functions (introduced in [13]). We prove several equalities involving trigonometric convolution products and distribution cosine functions as well as some relations between distribution cosine functions and ultradistribution semigroups.

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

In this work, we state and prove the sine, cosine I, cosine II, sine-cosine and cotangent rules for spherical triangles on the hyperbolic unit sphere H02 in the Lorentzian space R13.

In this note, using the multisection series method, we establish the formulas for various power sums of sine or cosine functions.

By use of the closure operation for an odd integer set, the minimal number of real multiplications needed to compute length-2m four-type discrete cosine and sine transforms over the field Q, of rational numbers has been demonstrated.

The product formula for the sine function is proved using the Gamma function and elementary probability theory. Some corollaries of the sine formula are also pointed out.

Let \(\{F_t : t \geq 0\}\) be a family of continuous additive set-valued functions defined on a convex cone \(K\) in a normed linear space \(X\) with nonempty convex compact values in \(X\). It is shown that (under some assumptions) a regular sine family associated with \(\{F_t : t \geq 0\}\) is continuous and \(\{F_t : t \geq 0\}\) is a continuous cosine family.