Chapter 9.2, Problem 53E

A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of P is (r, 0), and if the parameter θ is chosen as in the figure, show that parametric equations of the involute are

$\begin{array}{cc}x=r\left(\cos \theta +\theta \sin \theta \right)& y=r\left(\sin \theta -\theta \cos \theta \right)\end{array}$