Textbook Problem

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 siring 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

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