Derivatives of Sine and Cosine

Calculus 1 · Axiom Academy

LESSON Derivatives of Sine & Cosine Watch the slope of the sine curve, sampled point by point, trace out the cosine curve — and see why cosine's slope is negative sine. 1. Sine and Cosine Live on the Unit Circle Every angle picks out one point on the unit circle. Its horizontal coordinate is and its vertical coordinate is . As the point sweeps around, watch each coordinate drop down to its axis — sine and cosine are just the shadows the point casts. x-shadow = cosine, y-shadow = sine 2. The Slope of Sine Is Cosine Here is the heart of it. Slide a point along and keep a tangent line glued to it. At each x , measure that line's slope and plot it as a new point on the graph below. Those slope-points don't scatter randomly — they trace out a curve, and that curve is exactly . The sine curve climbs steeply near x=0 (slope 1 ), levels off at its peak (slope 0 ), then falls. Plot that climbing-then-falling slope and you've drawn one full cosine wave — leading to the result: 3. Check It at the Peaks and Zeros If is right, it has to survive a spot-check at the landmark angles. Watch the tangent jump to each one — the number on the tangent (its slope) should land exactly on every time. The curve is rising as steeply as it ever does — slope =1 , matching . At the very top the curve is momentarily flat — slope =0 , matching . The curve is falling at its steepest — slope =-1 , matching . Flat again at the bottom — slope =0 , matching . 4. The Slope of Cosine Is Negative Sine

This is the written version of the interactive lesson above. See the full Calculus 1 course.