We explore a pair of matrix solutions to a certain discrete system which has various properties similar to the familiar continuous trigonometric functions, including basic identities and sum and difference of two angles formulas. Then we examine separation properties of these matrices. An oscillation result is also given.

This entry provides an analytic proof to Ptolemy’s Theorem using polar form transformation and trigonometric identities. In this formalization, we use ideas from John Harrison’s HOL Light formalization [1] and the proof sketch on the Wikipedia entry of Ptolemy’s Theorem [3]. This theorem is the 95th theorem of the Top 100 Theorems list [2].

Properties of the q-ultraspherical polynomials for q being a primitive root of unity are derived using a formalism of the soq(3) algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of q-beta integrals of Ramanujan. Mathematics Subject Classifications (1991). 17B37, 33D80

We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Laĺın. As a corollary, we also obtain three q-series expansions for the dilogarithm.

The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of infinitesimal angles, these identities lead to the power series expansions of the trigonometric ...

HIGHLIGHTS This work introduces the Gudermannian function as a sigmoid growth model. Hyperbolic and trigonometric identities provide several parameterizations of this The model can be good alternative to classical models.

In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ Lambert series identities. Many people have attempted confirming some of those identities the Gosper’s list, mainly by using $q$-trigonometric this paper we either prove or disprove all list S. Ramanujan’s theta function N. Bailey’s summation formula. process, obt...

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, the Gauss circle problem Dirichlet divisor problem, in an earlier paper, three of present authors derived representations for certain sums products trigonometric functions as double series Bessel [8] . These are generalized paper introducing novel notion balanced derivatives, leading to further the...