(The material in this chapter is based on the presentation in [ 11 ].

Consider once again the point $P=(r\,\cos\theta,r\,\sin\theta)$ on the circle of radius $r$, as shown in Figure 3.2, and recall the geometric definition of the basic trigonometric functions in (Equations (1) and (2) of §3.2)

We would like to compute the derivatives of these functions. What do we know? We know that (infinitesimal) arclength along the circle is given by \begin{equation} ds = r\,d\theta \end{equation} but we also have the (infinitesimal) Pythagorean Theorem, which tells us that \begin{equation} ds^2 = dx^2 + dy^2 \end{equation} Furthermore, from $x^2+y^2=r^2$, we obtain \begin{equation} x\,dx + y\,dy = 0 \label{rdiff} \end{equation} Putting this information together, we have \begin{equation} r^2\,d\theta^2 = dx^2 + dy^2 = dx^2 \left( 1 + \frac{x^2}{y^2} \right) = r^2\,\frac{dx^2}{y^2} \end{equation} so that \begin{equation} d\theta^2 = \frac{dx^2}{y^2} = \frac{dy^2}{x^2} \end{equation} where we have used (\ref{rdiff}) in the last step. Carefully using Figure 3.2 to check signs, we can take the square root and rearrange terms to obtain \begin{eqnarray} dy &=& x\,d\theta \nonumber\\ dx &=& -y\,d\theta \end{eqnarray} Finally, (Equations (1) and (2) of §3.2) and using the fact that $r=\hbox{constant}$, we recover the familiar expressions \begin{eqnarray} d\sin\theta &=& \cos\theta\,d\theta \nonumber\\ d\cos\theta &=& -\sin\theta\,d\theta \end{eqnarray} We have thus determined the derivatives of the basic trigonometric functions from little more than the geometric definition of those functions and the Pythagorean Theorem — and the ability to differentiate simple polynomials.