Sine and Cosine Waves

Algebra 2 · Axiom Academy

Watch a point circle the unit circle and its coordinates unroll into the sine and cosine waves — two curves, one quarter-turn apart. Send a point around the unit circle. At every angle its height above the center is exactly . Plot that height against , and as the angle sweeps from 0 to the heights unroll to the right into the sine curve. Where the point sits on the unit circle Same sweep, but now track the point's width — its x -coordinate, . Stand that horizontal length up as a height and carry it across, and the widths unroll into the cosine curve. Because the point starts at the far right, , so the cosine wave begins at its peak instead of at zero. Now read the width instead of the height The cosine wave starts at 1, not 0 One full trip around the circle is one period . Over that single cycle, 0 to , both waves share the same vertical shape — and they trade off: where one is at a peak or trough, the other is crossing zero. Zeros at . Maximum of 1 at . Minimum of -1 at . Maximum of 1 at x=0 and . Zeros at . Minimum of -1 at . 4. Same Wave, a Quarter-Turn Apart Sine and cosine are not two different shapes — they are the same wave , offset by a quarter-turn. Slide the sine curve left by and it lands exactly on the cosine curve. Cosine is sine shifted left by a quarter-turn Equivalently, sine is cosine shifted right You built both waves straight from the unit circle — sine from the height, cosine from the width — and saw they are one shape, a quarter-turn apart.

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