<![CDATA[This paper presents an innovative extension of trigonometric functions to parabolic geometry, introducing the parabolic sine (sinp u) and parabolic cosine (cosp u) functions. Geometrically, sinp u and cosp u are defined via the relationship between a point on a parabola and its focus: sinp u represents the vertical displacement ratio, while cosp u corresponds to the horizontal displacement ratio, normalized by the focal distance. These functions generalize circular trigonometry to a parabolic framework, preserving key structural identities while exhibiting unique behaviors, such as fixed asymptotic values under angle variation. The objective of this study is to establish a rigorous foundation for parabolic trigonometry, derive its core identities, and demonstrate its applicability. Using a geometric-analytic approach, we redefine trigonometric concepts via parabola-centric constructions, adapt Euler’s formula to parabolic segments, and derive exponential representations of sinp u and cosp u. This method leverages differential geometry and algebraic invariance to ensure consistency with classical trigonometry while extending its scope. Key results include: (1) Proofs of sinp u, and cosp u; (2) Exponential forms: sinp u, and cosp u; (3) As the parabolic imaginary unit. Unlike circular trigonometry adaptations, our approach provides intrinsic geometric consistency with parabolic functions, enabling exact solutions for parabolic arc lengths and focal properties. This contrasts with numerical or linearized methods that sacrifice accuracy for simplicity. Theoretically, unifies parabolic geometry with analytic trigonometry, opening pathways for conic-section-generalized trigonometry, enhancing modeling in optics (parabolic mirrors), structural engineering (cable-supported arches), and ballistics (trajectory optimization), offering a novel pedagogical tool to bridge classical and modern geometry.]]>
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