Sine and Cosine Graphs

Pre-Calculus · Axiom Academy

Where the two most important waves in mathematics come from, and the simple grid of key points that lets you draw either one by hand. Put a point on the unit circle and let it travel counterclockwise at a steady speed, sweeping out the angle x . Its height above the center is, by definition, . Carry that height to the right at each instant and the moving point paints the sine curve. A point on the unit circle at angle x Same point, same circle — but now read its horizontal distance from the center instead of its height. That distance is . Unroll it to the right the same way and you get the cosine curve, which starts at its peak : at x=0 the point sits all the way to the right, so . The width (the x-coordinate) IS the cosine wave Height gives you sine; width gives you cosine. Both come from one point on one circle — that shared origin is the reason the two graphs are so closely related, as the next steps make precise. 3. Five Key Points, Every Quarter Turn You never need a table of values to sketch these curves. Both have period and amplitude 1 about the midline y=0 , so each one is completely pinned down by what it does at the quarter marks . Sine goes zero–max–zero–min–zero ; cosine goes max–zero–min–zero–max . Look again at those two rows of key points: cosine's behavior is sine's behavior read a quarter period early . That's not a coincidence — it's an identity. Slide the entire sine wave to the left by and it lands exactly on the cosine wave.

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