An elementary treatise on elliptic functions as trigonometry

Keywords: Elliptic functions, Circular functions, Geometric characteristics, Trigonometric identifies


This article concerns the examination of trigonometric identities from an elliptic perspective. The treatment of elliptic functions presented herein adheres to a structure analogous to the traditional exposition of trigonometric functions, with the exception that an ellipse replaces the unit circle. The degree of similarity between the elliptic functions and their trigonometric counterparts is moderated by the periodicity of the so-called El- functions. These identities not only establish the values of the functions, but also establish a correlation between their ratios and the major and minor axes of the underlying ellipse. The resemblance between the functions is somewhat modified by the periodic nature of the El-identities, whereby each ratio is associated with the major and minor axis of the ellipse. This article adopts the notation (E) to denote the El- functions and distinguish them from the opposite circular functions.


Download data is not yet available.


Al-ossmi, L. (2023). Elementary treatise on melted-like curves derived from center ellipse and circle. Iraqi Journal For Computer Science and Mathematics, 4(1), 114–129.

Al-ossmi, L. H. M. (2022). Nada’s curve towards a new curvature produced by the tangent of a circle and an ellipse: The nada’s curve. Iraqi Journal For Computer Science and Mathematics, 4(1), 1–9.

Alías, F., Socoró, J. C., & Sevillano, X. (2016). A review of physical and perceptual feature extraction techniques for speech, music and environmental sounds. Applied Sciences, 6(5), 143.

Alquran, M., & Jarrah, A. (2019). Jacobi elliptic function solutions for a two-mode KdV equation. Journal of King Saud University-Science, 31(4), 485–489.

Altman, R., & Kidron, I. (2016). Constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle. International Journal of Mathematical Education in Science and Technology, 47(7), 1048–1060.

Bialy, M. (2022). Mather β-Function for ellipses and rigidity. Entropy, 24(11), 1600.

Boyer, C. B., & Merzbach, U. C. (1968). A history of mathematics john wiley & sons. Inc New York, London, Sydney.

Byrd, P. F., & Friedman, M. D. (1971). Handbook of elliptic integrals for engineers and scientists. Berlin, Heidelberg.

Dimitrov, B. G. (2022). Elliptic integrals for calculation of the propagation time of a signal, emitted and received by satellites on one orbit. AIP Conference Proceedings, 2522(1), 90004.

Dragović, V., & Radnović, M. (2011). Poncelet porisms and beyond: Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Springer Science & Business Media.

Faulkner, B., Johnson‐Glauch, N., San Choi, D., & Herman, G. L. (2020). When am I ever going to use this? An investigation of the calculus content of core engineering courses. Journal of Engineering Education, 109(3), 402–423.

Glassmeyer, D., Brakoniecki, A., & M. Amador, J. (2019). Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship. International Journal of Mathematical Education in Science and Technology, 50(4), 527–556.

Hazewinkel, M. (2001). Jacobi elliptic functions. Encyclopedia of Mathematics, Springer.

Hery, C., & Ramamoorthi, R. (2012). Importance sampling of reflection from hair fibers. Journal of Computer Graphics Techniques (JCGT), 1(1), 1–17.

Kim, D., & Kim, Y. (2021). Causal image retrieval with ellipse-based object proposal. Journal of Visual Communication and Image Representation, 75(1), 102855.

Li, Y., Wu, L., & Li, L. (2021). Research on a new multiple criteria decision making method based on the ellipse membership function. Journal of Intelligent & Fuzzy Systems, 41(1), 1357–1371.

Liu, J., Shao, Y., & Lim, T. C. (2012). Vibration analysis of ball bearings with a localized defect applying piecewise response function. Mechanism and Machine Theory, 56, 156–169.

Liu, Q., & Gao, Y. (2021). The optimal approximation of the eccentricity of the ellipse with constrained arc length. Journal of Applied Mathematics and Computing, 68(12), 653–669.

Liu, S., Fu, Z., Liu, S., & Zhao, Q. (2001). Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A, 289(1–2), 69–74.

Liu, S. S., Fu, Z. T., & D., L. S. (2001). Expansion method about the jacobi elliptic function and its applications to nonlinearwave E2 quations. Acta Physici Sinica (in Chinese), 50(11), 2068–2073.

Reznik, D., Garcia, R., & Koiller, J. (2021). Fifty new invariants of n-periodics in the elliptic billiard. Arnold Mathematical Journal, 7(1), 341–355.

Salas, A. H., Abu Hammad, M., Alotaibi, B. M., El-Sherif, L. S., & El-Tantawy, S. A. (2022). Closed-form solutions to a forced damped rotational pendulum oscillator. Mathematics, 10(21), 4000.

Schaum, A. (2009). Advanced hyperspectral detection based on elliptically contoured distribution models and operator feedback. 2009 IEEE Applied Imagery Pattern Recognition Workshop (AIPR 2009), 1–5.

Shen, M., Wang, J., Joseph, A., Peng, F. Z., Tolbert, L. M., & Adams, D. J. (2006). Constant boost control of the z-source inverter to minimize current ripple and voltage stress. IEEE Transactions on Industry Applications, 42(3), 770–778.

Siyepu, S. W. (2015). Analysis of errors in derivatives of trigonometric functions. International Journal of Stem Education, 2, 1–16.

Smart, J. R., & Schwandt, E. A. (2012). Elliptic modular functions: An introduction (Vol. 203). Springer Science & Business Media.

Sterling, M. J. (2023). Trigonometry for dummies. John Wiley & Sons.

Törner, G., Potari, D., & Zachariades, T. (2014). Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts. ZDM, 46, 549–560.

Turner, H. R. (2010). Science in medieval Islam: An illustrated introduction. University of Texas Press.

Wang, B., & Jin, G. (2022). Solving a class of nonlinear evolution equations using jacobi elliptic functions. Journal of Physics: Conference Series, 2381(1), 12038.

Wang, P., Fu, Y., Zhang, J., Li, X., & Lin, D. (2018). Learning urban community structures: A collective embedding perspective with periodic spatial-temporal mobility graphs. ACM Transactions on Intelligent Systems and Technology (TIST), 9(6), 1–28.

Wang, W., Stuijk, S., & De Haan, G. (2015). A novel algorithm for remote photoplethysmography: Spatial subspace rotation. IEEE Transactions on Biomedical Engineering, 63(9), 1974–1984.

Yang, C., Wang, J., & Liu, X. (2021). An efficient nonlocal algorithm for ellipse detection. Signal Processing, 187(1), 108127.

How to Cite
Al-ossmi, L. H. M. (2023). An elementary treatise on elliptic functions as trigonometry. Alifmatika: Jurnal Pendidikan Dan Pembelajaran Matematika, 5(1), 1-20.
Abstract viewed = 192 times
FULL TEXT PDF downloaded = 277 times