# Calculus-Free Derivatives of Sine and Cosine

Calculus-Free Derivatives of Sine and Cosine

As a large square of side curls into a cylinder, an uncurled unit square is kept tangent to it, so that the unit square's diagonal always lies tangent to the helix formed by the larger square's diagonal. Projecting the helix into three mutually perpendicular planes yields a sinusoid, a "cosinusoid", and a circle; projecting the unit square's diagonal yields segments tangent to those curves. Basic geometric analysis of this figure then provides a straightforward development of the formulas for the derivatives of the sine and cosine functions from calculus… with nary a difference quotient to be seen!

2π