jmc

We have that $$\begin{array}{rl}\sin 60^{\circ }& =\cos 30^{\circ },\\ \cos 42^{\circ }& =\cos (360^{\circ }-42^{\circ })=\cos 318^{\circ },\\ -\sin 12^{\circ }& =-\cos (90^{\circ }-12^{\circ })=-\cos 78^{\circ }=\cos (180^{\circ }-78^{\circ })=\cos 102^{\circ },\\ -\cos 6^{\circ }& =\cos (180^{\circ }-6^{\circ })=\cos 174^{\circ },\end{array}$$so $$\cos \theta =\cos 30^{\circ }+\cos 318^{\circ }+\cos 102^{\circ }+\cos 174^{\circ }.$$If we plot $(\cos t,\sin t)$ for $t=30^{\circ },$ $102^{\circ },$ $174^{\circ },$ $246^{\circ },$ and $318^{\circ },$ the five points form the vertices of a regular pentagon.



Then by symmetry, the sum of the $x$-coordinates is $$\cos 30^{\circ }+\cos 102^{\circ }+\cos 174^{\circ }+\cos 246^{\circ }+\cos 318^{\circ }=0.$$Thus, $$\begin{array}{rl}\cos \theta & =-\cos 246^{\circ }\\ & =-\cos (360^{\circ }-246^{\circ })\\ & =-\cos 114^{\circ }\\ & =\cos (180^{\circ }-114^{\circ })\\ & =\cos 66^{\circ }.\end{array}$$Thus, the smallest such $\theta$ is $\boxed{66^{\circ }}.$