This paper develops a comprehensive meta-operational framework for trigonometric and inverse trigonometric functions, elevating them to the status of mutually inverse operational pairs fully integrated into the meta-operational mathematics established in previous works. We define sine, cosine, tangent and their inverses as elements of the operation space C = Op^ (C) and verify that they satisfy the ten axioms of meta-operational mathematics. The algebraic structure generated by these operations forms a Hopf sub-operad Pₓₑ₈₆ of the endomorphism operad End (C), whose coproduct encodes the classical addition formulas. We construct explicit Hopf algebra morphisms to the Connes--Kreimer renormalization Hopf algebra, thereby embedding sine-Gordon type quantum field theories into the meta-operationaln framework. Bornological convergence is employed to rigorously handle infinite series expansions (Taylor series, Euler products, continued fractions) of trigonometric and inverse trigonometric operations. The path integral trace is reinterpreted as a co-unit on this operad, and spectral triples with trigonometric potentials are shown to be stable under Bornological limits. All classical trigonometric identities are rewritten as meta-operational equalities, and the self-action dynamics of sine and cosine operations are analyzed in terms of fixed points and Julia sets. Furthermore, we categorify the framework to a 2-category and outline -categorical extensions. Numerical algorithms (truncated exponential approximation, continued fraction evaluation) are provided with rigorous Bornological error bounds. This work places the trigonometric--inverse trigonometric mutually inverse pair on an equal theoretical footing with the exponential--logarithm pair in meta-operational mathematics, providing a unified algebraic language for periodicity, compact symmetry, and renormalization.
shifa liu (Wed,) studied this question.