\(\Leftrightarrow cosx-cos7x-3\sqrt{3}sinx=0\)
\(\Leftrightarrow2sin4x.sin3x-3\sqrt{3}sinx=0\)
\(\Leftrightarrow2sin4x.\left(3sinx-4sin^3x\right)-3\sqrt{3}sinx=0\)
\(\Leftrightarrow sinx\left(6sin4x-8sin^2x.sin4x-3\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\Rightarrow x=...\\6sin4x-8sin^2x.sin4x=3\sqrt{3}\left(1\right)\end{matrix}\right.\)
Xét \(\left(1\right)\Leftrightarrow6sin4x-4sin4x\left(1-cos2x\right)=3\sqrt{3}\)
\(\Leftrightarrow2sin4x+4sin4x.cos2x=3\sqrt{3}\)
\(\Leftrightarrow sin4x+4sin2x.cos^22x=\frac{3\sqrt{3}}{2}\)
Ta có:
\(1=sin^22x+\frac{cos^22x}{2}+\frac{cos^22x}{2}\ge3\sqrt[3]{\frac{\left(sin2x.cos^22x\right)^2}{4}}\)
\(\Rightarrow\left(sin2x.cos^22x\right)^2\le\frac{4}{27}\Rightarrow sin2x.cos^22x\le\frac{2\sqrt{3}}{9}\)
\(\Rightarrow sin4x+4sin2x.cos^22x\le1+\frac{8\sqrt{3}}{9}< \frac{3\sqrt{3}}{2}\) nên pt vô nghiệm
