Unit Circle Definition

Algebra 2 · Axiom Academy

Sine and cosine aren't mysterious — they're just the coordinates of a point traveling around a circle of radius one. 1. Sine and Cosine Are Coordinates Send a ray out from the origin at angle , measured counter-clockwise from the positive x -axis. It crosses the unit circle at exactly one point. The first coordinate of that point is , and the second is — nothing more exotic than an x and a y . For instance, at the ray lands on the diagonal, where . The point where the ray meets the circle So it always sits on the circle x^2+y^2=1 Because and are just the x - and y -coordinates, their signs are simply the signs of x and y where the point sits. On the left half of the plane , so is negative; on the bottom half , so is negative. Now let keep increasing and record the point's height — its y -coordinate, — against on the graph to the right. That rising and falling height traces the sine wave . Do the same with the horizontal coordinate and you get the cosine wave . The point's height, plotted against One full turn repeats the pattern — period The cleanest values happen at the four quadrantal angles , where the ray lands exactly on an axis. There the point sits on a number line, so one coordinate is and the other is 0 . ( 0 , −1 ) · bottom of the circle The famous first-quadrant angles are read off the same way — each point's x is and its y is : Notice and swap x and y — they are mirror images across the diagonal.

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