New Year's Countdown Day 20: Sine $\pm$ Cosine

$\begin{aligned} \tan(20x^\circ) = \frac {\sin 17^\circ + \cos 17^\circ}{\sin 17^\circ - \cos 17^\circ} & = \frac {\sqrt 2 \left(\frac 1{\sqrt 2} \sin 17^\circ + \frac 1{\sqrt 2} \cos 17^\circ\right)}{-\sqrt 2 \left(\frac 1{\sqrt 2} \cos 17^\circ - \frac 1{\sqrt 2} \sin 17^\circ\right)} = - \frac {\sin 17^\circ \cos 45^\circ +\cos 17^\circ \sin 45^\circ}{\cos 17^\circ \cos 45^\circ - \sin 17^\circ \sin 45^\circ} = - \frac {\sin 62^\circ}{\cos 62^\circ} = -\tan 62^\circ \end{aligned}$

The smallest $|x|$ comes from $20x = - 62^\circ\implies |x| = 3.1 = \dfrac {31}{10}$ $\implies p+q = 31+10 = \boxed{41}$ .