Unveiling Janous's Conjecture: Calculus, Algebraic, and Computational Approaches

We give a proof of Janous's conjecture: 2 < \frac{\sin A}{A} + \frac{\sin B}{B} + \frac{\sin C}{C} \leq \frac{9\sqrt{3}}{2\pi}, where A, B, C > 0 and A + B + C = \pi which combines integral calculus, geometric optimization, symmetry arguments, supplemented by computational techniques. We also provide an algebraic proof avoiding calculus.