Tìm số nghiệm có dạng m π 3 , m ∈ Z trên đoạn [ 0 ; 2 π ] của phương trình 1 + s i n x + c o s x + s i n 2 x + c o s 2 x = 0

\[{\rm{1 + sinx + cosx + sin2x + cos2x = 0}}\]

\[ \Leftrightarrow \left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{x + si}}{{\rm{n}}^{\rm{2}}}{\rm{x + 2sinx}}{\rm{.cosx}}} \right){\rm{ + }}\left( {{\rm{sinx + cosx}}} \right){\rm{ + }}\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{x}} - {\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{x}}} \right){\rm{ = 0}}\]

\[ \Leftrightarrow {\left( {{\rm{sinx + cosx}}} \right)^{\rm{2}}}{\rm{ + }}\left( {{\rm{sinx + cosx}}} \right){\rm{ + }}\left( {{\rm{cosx}} - {\rm{sinx}}} \right){\rm{.}}\left( {{\rm{sinx + cosx}}} \right){\rm{ = 0}}\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{\rm{sinx + cosx = 0}}}\\{{\rm{2cosx + 1 = 0}}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{sin\left( {x + \frac{\pi }{4}} \right){\rm{ = }}0}\\{cosx\,{\rm{ = }} - \frac{1}{2}}\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x{\rm{ = }} - \frac{\pi }{4} + k\pi }\\{x{\rm{ = }} \pm \frac{{2\pi }}{3} + k2\pi }\end{array}} \right.,k \in \mathbb{Z}\)

Vì\[{\rm{x}} \in \left[ {{\rm{0; 2\pi }}} \right] \Rightarrow {\rm{x}} \in \left\{ {\frac{{{\rm{3\pi }}}}{{\rm{4}}}{\rm{; }}\frac{{{\rm{7\pi }}}}{{\rm{4}}}{\rm{; }}\frac{{{\rm{2\pi }}}}{{\rm{3}}}{\rm{; }}\frac{{{\rm{4\pi }}}}{{\rm{3}}}} \right\}\]

Đáp án cần chọn là: C