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All Journal Alifmatika: Jurnal Pendidikan dan Pembelajaran Matematika
Al-ossmi, Laith H. M.
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Beyond circular trigonometry: Parabolic functions from geometric identities Al-ossmi, Laith H. M.
Alifmatika (Jurnal pendidikan dan pembelajaran Matematika) Vol 7 No 1 (2025): Alifmatika - June
Publisher : Fakultas Tarbiyah Universitas Ibrahimy

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35316/alifmatika.2025.v7i1.1-33

Abstract

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.
The high-accuracy geometric approximation of the ellipse's perimeter by the measuring right-angled triangle Al-ossmi, Laith H. M.
Alifmatika (Jurnal pendidikan dan pembelajaran Matematika) Vol 7 No 2 (2025): Alifmatika - December
Publisher : Fakultas Tarbiyah Universitas Ibrahimy

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.35316/alifmatika.2025.v7i2.310-331

Abstract

This article confronts the persistent challenge of determining the exact perimeter of an ellipse. It proposes a high-accuracy geometric approximation centred on a uniquely defined Measuring Right-Angled Triangle (MRAT). Constructed with specific spatial and angular properties, the MRAT is positioned at a distance of 2b/π from the centre of a reference circle and terminates at its circumference at a 45° angle. The ellipse's center is co-located with the circle's center. The resulting values were rigorously compared against classical Ramanujan approximations, and the PRI test and high-precision graphical analysis were used to confirm significant accuracy. This high-accuracy geometric approximation method offers a computationally efficient alternative to traditional algebraic methods, enhancing both theoretical understanding and applied precision.
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