Unit Circle Exact Values: Geometric Construction for Key Angles

This example demonstrates how key angles on the unit circle correspond to exact coordinates and trigonometric values by using geometric relationships (including right triangles and similarity). It centers on the unit circle, the link between points on the circle and sine/cosine values, angle measurement/labeling, and similarity-based proportional reasoning to obtain exact results. Construction highlights include using Intersect to compute circle–line intersections that define points on the circle, Point to mark critical and auxiliary vertices, Circle to create the relevant (unit) circle, and Segment to draw triangle sides and structural edges. Distances are measured with Distance to extract the needed lengths, while Angle is used to construct/verify the target angles. Vector and Polygon help organize directed relations and the polygonal structure used for the similarity setup, and text labels (e.g., for angles and numeric markers such as α, d) present the exact values directly on the diagram. Suitable for high-school mathematics and early trigonometry instruction, especially for visualizing why special-angle values on the unit circle are exact rather than approximate.