This paper systematically extends the foundational framework of Operational Mathematics—originally developed for arithmetic hyperoperations—to the realm of trigonometric and inverse trigonometric functions. The core principle remains the extension of the repetition count of a fundamental operation from natural numbers to integers, rationals, reals, and complex numbers. Here the fundamental operation is chosen to be the sine function (with cosine and tangent obtained via elementary transformations). We establish a complete axiomatic system tailored to the unique properties of trigonometric functions: periodicity, boundedness, the presence of a parabolic real fixed point at the origin, and an infinite lattice of complex fixed points with varying multipliers. Rigorous constructions of integer-order, fractional-order, real-order, and complex-order iterates of sine and arcsine are given, with existence proved via Abel’s equation (for the parabolic fixed point) and Schröder’s equation (for hyperbolic complex fixed points). Uniqueness theorems under appropriate regularity conditions are established. The singularity structure of complex-order trigonometric iterates is deeply analyzed, and the associated Riemann surfaces are characterized. Furthermore, we prove that the fractional calculus of trigonometric operators and the fractional calculus of variations involving trigonometric derivatives are special cases of this extended Operational Mathematics in the continuous setting. The profound duality between the mathematics of angles (arguments) and the mathematics of trigonometric iterations is revealed and formalized via category theory. All constructions are accompanied by rigorous, step-by-step proofs—each theorem is provided with at least four detailed proof steps, and crucial theorems with at least eight. In particular, we transform previously stated open problems and conjectures into fully proved theorems by supplying complete derivations. A comprehensive framework for high-precision numerical computation is developed, including explicit error bounds and convergence analysis. This work demonstrates that Operational Mathematics is a universal language for iteration, encompassing not only algebraic operations but also the transcendental functions that lie at the heart of analysis.
Liu S (Wed,) studied this question.