Trigonometric Inequality: -3/2 ≤ cosα + cosβ + cosγ ≤ 3

This GeoGebra applet demonstrates the geometric interpretation and optimization of the trigonometric inequality -3/2 ≤ cosα + cosβ + cosγ ≤ 3. Through interactive sliders that control the angles α, β, and γ, users can observe the real-time computation and dynamic visualization of the sum cosα + cosβ + cosγ as each angle varies. The applet employs vector constructions and geometric diagrams to build an animated model that illustrates exactly when the expression reaches its maximum value of 3 and its minimum value of -3/2. When all three angles are equal to 0, the sum achieves its upper bound of 3; when the angles are arranged symmetrically at specific configurations related to the cube roots of unity on the unit circle, the sum reaches its lower bound of -3/2. This visual approach helps students develop an intuitive understanding of why the boundedness of trigonometric sums holds and under what conditions equality is attained. The interactive nature of the applet allows learners to experiment freely, dragging sliders to explore how small perturbations in one angle affect the overall sum, thereby reinforcing the connection between algebraic inequalities and their underlying geometric structure. Educators can use this tool in classroom settings to turn an abstract proof-based result into a hands-on discovery experience, making it particularly valuable for precalculus and calculus courses covering trigonometry, optimization, and inequality proofs.