phương trình <=> $5+ 3\sqrt{2}\sin x- \sqrt{2}\cos x -2\sin 2x=0$<=>$2(\sin x-\cos x)^{2}+\sqrt{2}(sin x-\cos x)+(2\sqrt{2}\sin x+3)=0$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})=0$$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2} \geq \frac{1}{4}$ và $\sin x\geq -1$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})\geq \frac{1}{4}+(2\sqrt{2}\sin x+\frac{11}{4})$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})> 0$Vậy phương trình vô nghiêm

phương trình <=> $5+ 3\sqrt{2}\sin x- \sqrt{2}\cos x -2\sin 2x=0$<=>$2(\sin x-\cos x)^{2}+\sqrt{2}(sin x-\cos x)+(2\sqrt{2}\sin x+3)=0$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})=0$$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2} \geq \frac{1}{4}$ và $\sin x\geq -1$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})\geq \frac{1}{4}+(2\sqrt{2}\sin x+\frac{11}{4})$=>$(\sqrt{2}(\sin x-\cos x)+\frac{1}{2})^{2}+(2\sqrt{2}\sin x+\frac{11}{4})> 0$Vậy phương trình vô nghiêm