A New Formula for Approximating the Sine Function As an Infinite Product

Overview

Approximating values of the sine function has challenged mathematicians for centuries, leading to a wide variety of formulas and techniques. Efficient approximations for trigonometric functions are extremely useful for applications where speed is paramount to accuracy, such as real time graphics rendering, large-scale physics simulations, and compute-limited robotics microcontrollers. In this project, I derive a new formula for approximating the values of the sine function. I explore the hypothesis that for polynomial representations of the sine function there exists an optimal expression for the prefactor that gives the best convergence. My approximation is very closely related to Euler’s infinite product for the sine function, but converges faster, while maintaining numerical accuracy. In Euler’s infinite product for sine, the sine function is written in a factored form through the zeros of the original function; using a similar analytical approach, my work derives a modified form for the prefactor that achieves the necessary error with the fewest number of terms. I perform an error analysis of the convergence of this new form, and compare against other approximations for the sine function.

Competition history

• CSEF 2026 Mathematical Sciences (Senior Division) · Entry S-14-02

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