Unit Circle Mastery

Calculus Readiness · Axiom Academy

One labelled circle holds every special-angle value you will reach for in calculus — read it, don't memorize it. 1. Sixteen Points, Read Straight Off the Circle Walk one lap counterclockwise. At each of the sixteen special angles a point lights up, and its coordinates ARE its values: the x -coordinate is , the y -coordinate is . Watch the running point — every label is just where it lands. the point on the circle at angle θ x is the cosine, y is the sine 2. One Point, Four Quadrants: Reference Angles You only ever memorize the first quadrant. Take the point at , with the acute reference angle tucked against the x -axis. Reflect it across the y -axis, then the origin, then the x -axis — the same triangle reappears in Q2, Q3, Q4 with only the signs of the coordinates flipping. The magnitude never changes. . Here . Both coordinates positive. lands in Q4, where its reference angle is . Sine is the y -coordinate and y is negative in Q4, so . 3. Which Sign in Which Quadrant Once the reference angle gives you the size, the quadrant alone fixes the sign. Cosine is the x -coordinate, so it is positive on the right half ( x>0 ); sine is the y -coordinate, so it is positive on the top half ( y>0 ). The sweep below stamps the sign pair onto each quadrant in turn. Cosine follows the x-axis: positive in Q1 and Q4 (the right half), negative in Q2 and Q3. Sine follows the y-axis: positive in Q1 and Q2 (the top half), negative in Q3 and Q4.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.