Sum of Sine and Cosine as a Single Sine Curve

This GeoGebra example demonstrates that any linear combination a·sin(x) + b·cos(x) can be rewritten as a single sine function c·sin(x + α), showing that the sum of sine and cosine curves remains a sine curve with adjusted amplitude and phase shift. Key mathematical concepts include trigonometric identities, the auxiliary angle formula, function transformations (amplitude scaling and horizontal translation), and periodic function superposition. The construction uses objects a, b, c, α and functions f, g, h, with auxiliary elements k, d, e to dynamically visualize coefficient sliders, component graphs, and the resulting composite curve. It is suitable for high school trigonometry, pre-calculus, and university introductory courses, serving teachers for classroom demonstration and students for independent exploration.