The Unit Circle

Pre-Calculus · Axiom Academy

A circle of radius 1 turns angles into coordinates — and that single idea is the engine behind sine and cosine. The unit circle is the circle with radius 1 centered at the origin (0,0) . The "unit" is the radius: every point on it is exactly 1 unit from the center. Watch the radius swing — its tip stays glued to the circle the whole way around, because that distance never changes. a circle of radius 1 centered at the origin Measure an angle counter-clockwise from the positive x -axis and drop the point where that ray meets the circle. Its coordinates aren't arbitrary: the horizontal position is and the vertical position is . The animation drops the blue shadow to the x -axis (that length is ) and the orange shadow to the y -axis (that length is ). This is where sine and cosine live Cosine and sine aren't separate machines you have to trust — they are literally the x and y shadows of a point walking around this circle. Read the point, read the trig values. As the point travels all the way around, it passes through four quadrants, and the signs of change in each one. The live tag flips the instant the point crosses an axis — cosine sign tracks the left/right side, sine sign tracks the top/bottom. A handful of first-quadrant angles show up constantly, and their coordinates are worth memorizing. They come from the and triangles. The animation snaps the radius to each in turn — — and lands the point on its exact coordinate.

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