Every Derivative of sin(x) Is Hiding on This Circle. Take the slope of sin x at every point and the slopes trace a new wave: cos x. It’s the same wave, a quarter turn ahead on the unit circle.
Do it again and you get −sin x, then −cos x. On the fourth derivative you land exactly back on sin x. Four derivatives, one full turn.
Why? Because d/dx sin(x) = sin(x + π/2). Each derivative shifts the wave by π/2, and four of those shifts add up to 2π. That’s a full rotation, so you’re back where you started.
One catch: this only works when x is in radians. In degrees, every derivative picks up an extra factor of π/180.
Every exact value on screen is checked against the true derivative to 40 digits.
🔊 Listen closely: each chime’s pitch is the slope value it marks, so the first pass literally plays the cosine.
#calculus #math #unitcircle
