Chapter 10.3, Problem 61ES

As illustrated in the accompanying figure, let $P\left(r,\theta \right)$ be a point on the polar curve $r=f\left(\theta \right),$ let $\psi$ be the smallest counterclockwise angle from the extended radius OP to the tangent line at P, and let $\varphi$ be the angle of inclination of the tangent line. Derive the formula

$\tan \psi =\frac{r}{dr/d\theta }$

by substituting tan $\varphi$ for $dy/dx$ in Formula (2) and applying the trigonometric identity

$\tan \left(\varphi -\theta \right)=\frac{\tan \varphi -\tan \theta }{1+\tan \varphi \tan \theta }$