How to Fill Out the Entire Unit Circle

Miacademy & MiaPrep Learning Channel

This educational video serves as a comprehensive guide to mastering the unit circle beyond the first quadrant. It begins by addressing the conceptual shift required to move from right-triangle trigonometry to circular functions, specifically tackling the problem of angles like 90 degrees where a physical triangle cannot exist. The narrator transitions definitions from "opposite/adjacent" to coordinate geometry (x, y), allowing students to find sine, cosine, and tangent values for any angle from 0 to 360 degrees. Key themes include mathematical symmetry, reflection, and coordinate geometry. The video demonstrates how to derive values for the second, third, and fourth quadrants by simply reflecting known values from the first quadrant across the x and y axes. It also explicitly addresses the concept of undefined values, such as the tangent of 90 and 270 degrees, explaining the mathematical reasoning behind division by zero in this context. For educators, this video is an invaluable tool for bridging the gap between Geometry and Pre-Calculus. It moves beyond rote memorization by teaching students a logical method to construct the unit circle from scratch using symmetry and critical thinking. The inclusion of guided practice problems at the end reinforces the application of these concepts, making it ideal for classroom instruction or independent review.