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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 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, 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.

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 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.

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 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.

In this note we give an elementary derivation for the power series expansions for the sine and the cosine function.

Questions involving the exact values of the trigonometric functions of 180, 360, 540 and 720 have been popular in compilations of challenging mathematical problems, and in the problems sections of mathematics journals. (See, for example, problems 189 and 339 in [1] and the sections entitled Trigonometry: numerical evaluation, Trigonometry: numerical identities and Trigonometry: theory of equations in [2].) Invariably, the determination of these values make little or no use of geometry; rather, the derivations rely heavily upon trigonometric identities, such as the double and triple angle formulas or the product-to-sum formulas. In this note, we will use a purely geometric approach to derive the values of the sine and cosine of multiples of 18?. Therefore, this capsule can be used as enrichment or investigative material for students in trigonometry or precalculus courses. Since 0? and 900 are already standard trigonometric angles, we consider only angles of measure 18?, 36?, 54? and 720. Using Figure 1, we first show that the sine and cosine of 18?, 36?, 540 and 72? can all be expressed as functions of x. Toward this end, draw the altitude CD from vertex C to the side AB (Figure(b)), and l n et BD = y. For the right triangle BCD, we find (by the Pythagorean theorem) that CD = x2 y2, and for the right triangle ADC, we see that CD = 1 (1 y)2 = 2y y2 Equating the expressions for CD leads to y = x2. Therefore

A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board. Expansion of the polynomial by De Moivre's theorem expresses the function as a sum of sine terms and a sum of cosine terms. The polynomial may then be graphed by the machine, and thereby the complex roots determined. A number of typical curves are given to illustrate the machine method of determining real, imaginary, and complex roots. An auxiliary method is described which gives all the real roots of the polynomial from a single graph.

(1987). The Derivatives of the Sine and Cosine Functions. The College Mathematics Journal: Vol. 18, No. 2, pp. 139-141.

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(1968). Sine and Cosine Integrals. Optica Acta: International Journal of Optics: Vol. 15, No. 2, pp. 195-196.

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.