Geometric construction of the tangent function on [-π/2, π/2]

This example demonstrates how the tangent function on the interval [-π/2, π/2] can be visualized through a geometric construction that links an angle α to an appropriate slope/ratio. It emphasizes the geometric meaning of tan(α), the relationship between angles and right-triangle/circle geometry, and how varying an angle produces a function behavior on a restricted domain. Construction is driven by GeoGebra commands indicated in the summary: Semicircle to set up the circular/semicircular scaffold; Point to create key points (e.g., A, B, C, D, and many auxiliary points); Intersect to compute intersection points that define the segment/line placements; If to select or adjust configurations depending on the angle; Segment and Ray to build the required lines whose geometry encodes the tangent relationship; and Angle to measure and control the driving angle α. Intended for high school and early university learners studying trigonometry and graphing via dynamic geometric insight, useful for classroom demonstrations and self-guided exploration.