Determining the Optimal Free Kick Position in Soccer to Maximize the Shooting Angle via Trigonometric, Calculus, and Geometric Methods

This investigation explores how trigonometry and geometry can determine the optimal free kick position along a soccer touchline to maximize the shooting angle towards the goal.

The researcher employs three distinct methodologies: algebraic optimization via the AM-GM inequality, calculus-based differentiation (using compound angle and arctangent identities), and circle geometry applying the Inscribed Angle Theorem. Results are cross-verified using data tables and Desmos graphing.

The study concludes that the optimal distance follows the formula y = √ab, and that the relationship forms a hyperbola. A key strength is the rigorous comparison between complex hyperbolic models and simplified asymptotic approximations, providing both mathematical depth and practical real-world applicability.