Unit Circle: Introduction

Calculus Readiness · Axiom Academy

One circle of radius 1, one angle sweeping around it — and every value of sine and cosine falls right out of it. Draw a circle of radius 1 centred at the origin. Pick a point on it, and a single number — the angle measured from the positive x -axis — pins it down. The beautiful part: that point's coordinates are exactly . So sine and cosine stop being mystery buttons on a calculator and become something you can simply read off the circle . Watch the arm sweep one full turn. The point traces the circle, and at every instant it drops two shadows — straight down onto the x -axis (that's ) and straight across onto the y -axis (that's ). The live coordinate is always . The point is — the two shadows it casts on the axes are exactly those coordinates. is the height, is the horizontal Drag the point around the circle. Drop a line straight down to the x -axis and you've built a right triangle whose hypotenuse is the radius 1. Its horizontal leg is and its vertical leg — the height — is . Because the hypotenuse is 1, the Pythagorean theorem becomes the identity . Move all the way round: the legs flip sign by quadrant, but never leaves 1. A radian is just the arc you swept Sweep the angle and watch the arc it carves along the circle. Now unroll that arc into a straight segment and lay it on a number line. Its length is the angle in radians — because on a circle of radius 1 the arc length is . A quarter turn is , a half turn is , and the whole way around is .

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