The two key functions of oscillation have specially neat derivatives: The slope of sin x is cos x ! And the slope of cos x is - sin x.

These come from one crucial fact: (sin x) / x approaches 1 at x = 0. This checks that the slope of sin x is cos 0 = 1 at the all-important point x = 0.

Professor Strang connects sine and cosine to moving around a circle,
or up and down for a spring, or in and out for your lungs.