Tìm số nghiệm thuộc đoạn [ 2 π ; 4 π ] của phương trình s i n 3 x c o s x + 1 = 0

\[\frac{{{\rm{sin3x}}}}{{{\rm{cosx + 1}}}}{\rm{ = 0}} \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{cosx \ne - 1}\\{sin3x = 0}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{x \ne \pi + k2\pi }\\{x = \frac{{k\pi }}{3}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \pm \frac{\pi }{3} + k\pi }\\{x = k2\pi }\end{array},k \in \mathbb{Z}} \right.\]

Vì \[{\rm{x}} \in \left[ {{\rm{2\pi ; 4\pi }}} \right] \Rightarrow 2{\rm{\pi }} \le {\rm{x}} \le 4{\rm{\pi }}\]

Xét:\[{\rm{2\pi }} \le \frac{{\rm{\pi }}}{{\rm{3}}}{\rm{ + k\pi }} \le {\rm{4\pi }} \Rightarrow \frac{{{\rm{5\pi }}}}{{\rm{3}}} \le {\rm{k\pi }} \le \frac{{{\rm{11\pi }}}}{{\rm{3}}} \Rightarrow \frac{5}{3} \le {\rm{k}} \le \frac{{11}}{3}\]

\[ \Rightarrow {\rm{k = 2; k = 3}} \Rightarrow {\rm{x = }}\frac{{{\rm{7\pi }}}}{{\rm{3}}}{\rm{; x = }}\frac{{{\rm{10\pi }}}}{{\rm{3}}}\]

Xét:\[{\rm{2\pi }} \le - \frac{{\rm{\pi }}}{{\rm{3}}}{\rm{ + k\pi }} \le {\rm{4\pi }} \Rightarrow \frac{{{\rm{7\pi }}}}{{\rm{3}}} \le {\rm{k\pi }} \le \frac{{{\rm{13\pi }}}}{{\rm{3}}} \Rightarrow \frac{7}{3} \le {\rm{k}} \le \frac{{13}}{3}\]

\[ \Rightarrow {\rm{k = 3; k = 4}} \Rightarrow {\rm{x = }}\frac{{{\rm{8\pi }}}}{{\rm{3}}}{\rm{; x = }}\frac{{{\rm{11\pi }}}}{{\rm{3}}}\]

Xét:\[{\rm{2\pi }} \le {\rm{k2\pi }} \le {\rm{4\pi }} \Rightarrow 1 \le {\rm{k}} \le 2 \Rightarrow {\rm{k = 1; k = 2}} \Rightarrow {\rm{x = 2\pi ; x = 4\pi }}\]

Đáp án cần chọn là: A