Geometric Construction of the Sine Function with Tangent and Symmetry

This example demonstrates how the sine function can be visualized through a geometry-based construction tied to a circle, showing how the graph emerges from an angle-driven mapping. It also highlights key behaviors such as tangent construction (tangent line direction/slope insight) and symmetry/periodicity patterns in trig functions. The main mathematical concepts include trigonometry (sine), the geometric meaning of function graphs, the idea of a tangent, and how symmetry and periodicity determine the overall shape. From the construction summary, it uses GeoGebra commands such as Point to define crucial points; OrthogonalLine and Segment to build perpendicular lines and connecting segments; Intersect to locate intersection points for correct positioning; Rotate and Circle to realize rotation-based mapping along the circle; Angle and CircleArc to measure and represent the driving angle and its corresponding arc; and Polygon to structure related geometric regions. Key objects (e.g., points like A, B, O, P, Q; curves/functions f and g including a sine-related object; and angle parameters such as α and ang) are used to link the geometry to the resulting function behavior. Suitable for middle school through early university preparation, especially for lessons on trigonometric modeling, graphing intuition, and geometric construction activities.