Inequality of the Sum of Squared Cosines of Triangle Angles

This applet demonstrates the inequality for the sum of squared cosines of a triangle's interior angles. For any triangle ABC, it satisfies 3/4 ≤ cos²A + cos²B + cos²C ≤ 3. Through dynamic geometric construction in GeoGebra, it visually illustrates the conditions and boundary cases, aiding the understanding of trigonometric functions in triangle geometry. Users can manipulate the vertices of triangle ABC interactively to observe how the sum cos²A + cos²B + cos²C changes as the triangle's shape varies. The applet dynamically displays the current value of the sum alongside the theoretical bounds of 3/4 and 3. The upper bound of 3 is achieved in the limiting case where the triangle degenerates — for instance, when all three angles approach zero, which corresponds to a collapsed triangle. In practice, as the triangle becomes increasingly degenerate, the sum approaches this upper limit. The lower bound of 3/4 is attained when the triangle is equilateral, where each angle equals 60 degrees and cos²(60°) = 1/4, yielding a sum of exactly 3/4. The applet allows students to verify this by dragging the vertices until the triangle approaches an equilateral configuration and observing the sum converge to 3/4. For a right triangle, the sum takes an intermediate value, further illustrating that the expression continuously varies between these two extremes depending on the triangle's shape. This hands-on exploration reinforces key connections between trigonometry and geometry, showing how algebraic inequalities have direct geometric interpretations. The interactive nature of the applet enables students to form and test conjectures, develop intuition about extremal cases, and deepen their understanding of why these particular bounds hold. By linking the abstract inequality to concrete visual feedback, this tool serves as an effective classroom resource for topics covering trigonometric identities, optimization in geometry, and the properties of triangles.