Show that the angle … between the ray from the origin and…

Show that the angle $\psi$ between the ray from the origin and the tangent line has $\tan \psi =F/(dF/d\theta )$.
Hint: If the tangent line is at an angle $\varphi$ with the horizontal, then $\tan \varphi$ is the slope $dy/dx$ in Problem 18. Therefore
$$\tan \psi =\tan (\varphi -\theta )=\frac{\tan \varphi -\tan \theta }{1+\tan \varphi \tan \theta }.$$
Substitute for $\tan \varphi$ and simplify like mad.

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §9.3 Slope, Length, and Area for Polar Curves, #20