Graphing Sine and Cosine

Calculus Readiness · Axiom Academy

LESSON Graphs of Sine and Cosine The two waves that power all of trigonometry — and they are nothing more than the unit circle, unrolled. 1. A Wave Is a Circle, Unrolled Send a point counterclockwise around the unit circle at angle . Its height above the center is exactly . Carry that height to the right, plotting it against , and it traces the sine wave — starting at 0 and rising. Watch the orange segment: the same height appears on the circle and on the wave at the same instant. A point on the unit circle at angle θ 2. Cosine Is the Same Circle, Read Sideways Now track the horizontal reach of the same point — its x -coordinate, which is . Stand that length up and unroll it to the right against , and you get the cosine wave . At the point sits at the far right, so the reach is a full 1 : cosine starts at its maximum and falls. It's the sine wave shifted left by . The point is at the far right: horizontal reach = 1 . Cosine begins at its maximum. The point is straight up: horizontal reach = 0 . The wave is crossing the midline, descending. The point is at the far left: reach = -1 . Cosine hits its minimum. Back to the far right: reach = 1 again. One period of cosine is complete. The same wave, half a step apart Sine is the height, cosine is the width — read off the very same spinning point. Because the height lags the width by a quarter turn, : cosine is just sine slid left by . 3. Five Points Sketch the Whole Wave

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