Trigonometric Functional Equations

Abstract

AbstractThe famous addition and subtraction rules for trigonometric functions can be represented by using functional equations. Some of these equations will be introduced and the stability problems for them will be surveyed. Section 12.1 deals with the superstability phenomenon of the cosine functional equation (12.1) which stands for an addition theorem of cosine function. Similarly, the superstability of the sine functional equation (12.3) is proved in Section 12.2. In Section 12.3, trigonometric functional equations (12.8) and (12.9) with two unknown functions will be discussed. It is very interesting that these functional equations for complex-valued functions defined on an amenable group are not superstable, but they are stable in the sense of Hyers and Ulam, whereas the equations (12.1) and (12.3) are superstable. In Section 12.4, we will deal with the Hyers–Ulam stability of the Butler–Rassias functional equation.KeywordsAbelian GroupFunctional EquationAmenable GroupTional EquationAddition TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.