Constructing the Tangent Function from the Unit Circle

This example demonstrates how to derive and visualize the tangent function using geometric relationships from the unit circle. It centers on the unit circle, angle-based constructions, right-triangle side ratios, and the meaning of tan(θ), linking the geometry to a functional representation. The construction uses GeoGebra commands Point, Angle, Segment, If, and Slope to create key points and angle measures, build the required segments, apply conditional logic (e.g., handling different angle regions or special cases), and compute/update the slope so that it reflects the tangent relationship as the angle changes. Visually and interactively, it relies on construction objects such as the main points (e.g., A, B, C, D), an angle parameter, and function/geometry elements (e.g., f, h and other auxiliary variables), along with controls like revcount and buttons (button1/button2) to drive the angle variation and update the resulting function/graph behavior. Suitable for high-school advanced learners and early university students to build intuition for trigonometric functions through geometric interpretation.