\(\begin{aligned} \sin \left(x-30^{\circ}\right) &=\cos 2 x \\ \cos \left(90^{\circ}-x+30^{\circ}\right) &=\cos 2 x \\ \cos \left(120^{\circ}-x\right) &=\cos 2 x \\ 120^{\circ}-x &=\left\{\begin{array}{l}2 x+n 360^{\circ} \\ 360^{\circ}-2 x+n 360^{\circ}\end{array}\right.\\ & \Rightarrow\left\{\begin{array}{l}3 x=120^{\circ}-n 360^{\circ} \\ x=-120+n 360^{\circ}\end{array}\right.\\ & \Rightarrow x=\left\{\begin{array}{l}40^{\circ}+n 120^{\circ} \\ 240^{\circ}+n 360^{\circ}\end{array}\right.\\ & \Rightarrow x=\left\{40^{\circ}, 160^{\circ}, 240^{\circ}, 280^{\circ}\right\}+n 360^{\circ} \end{aligned}\)