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.

Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.