Cho pt : 2 cos2x + sqrt(3) = 0 a) Tìm các nghiệm thuộc [ 0 ; 3π ] b) Tính tổng các nghiệm thuộc [ -π/2 ; 3π ]

Đáp án:

\(\eqalign{ & a)\,\,x \in \left\{ {{{5\pi } \over {12}};{{17\pi } \over {12}};{{29\pi } \over {12}};{{7\pi } \over {12}};{{19\pi } \over {12}};{{31\pi } \over {12}}} \right\} \cr & b)\,\,\,{{103\pi } \over {12}} \cr} \)

Giải thích các bước giải:

\(\eqalign{ & 2\cos 2x + \sqrt 3 = 0 \Leftrightarrow \cos 2x = - {{\sqrt 3 } \over 2} \cr & \Leftrightarrow \left[ \matrix{ 2x = {{5\pi } \over 6} + k2\pi \hfill \cr 2x = - {{5\pi } \over 6} + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{ x = {{5\pi } \over {12}} + k\pi \hfill \cr x = - {{5\pi } \over {12}} + k\pi \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr & a)\,\,x \in \left[ {0;3\pi } \right] \cr & + )\,\,0 \le {{5\pi } \over {12}} + k\pi \le 3\pi \cr & \Leftrightarrow 0 \le {5 \over {12}} + k \le 3 \cr & \Leftrightarrow - {5 \over {12}} \le k \le {{31} \over {12}} \cr & Ma\,\,k \in Z \Rightarrow k \in \left\{ {0;1;2} \right\} \cr & \Rightarrow x \in \left\{ {{{5\pi } \over {12}};{{17\pi } \over {12}};{{29\pi } \over {12}}} \right\} \cr & + )\,\,0 \le - {{5\pi } \over {12}} + k\pi \le 3\pi \cr & \Leftrightarrow 0 \le {{ - 5} \over {12}} + k \le 3 \cr & \Leftrightarrow {5 \over {12}} \le k \le {{41} \over {12}} \cr & Ma\,\,k \in Z \Rightarrow k \in \left\{ {1;2;3} \right\} \cr & \Rightarrow x \in \left\{ {{{7\pi } \over {12}};{{19\pi } \over {12}};{{31\pi } \over {12}}} \right\} \cr & b)\,\, + )\,\, - {\pi \over 2} \le {{5\pi } \over {12}} + k\pi \le 3\pi \cr & \Leftrightarrow - {1 \over 2} \le {5 \over {12}} + k \le 3 \cr & \Leftrightarrow - {{11} \over {12}} \le k \le {{31} \over {12}} \cr & Ma\,\,k \in Z \Rightarrow k \in \left\{ {0;1;2} \right\} \cr & \Rightarrow x \in \left\{ {{{5\pi } \over {12}};{{17\pi } \over {12}};{{29\pi } \over {12}}} \right\} \cr & + )\,\, + )\,\, - {\pi \over 2} \le - {{5\pi } \over {12}} + k\pi \le 3\pi \cr & \Leftrightarrow - {1 \over 2} \le {{ - 5} \over {12}} + k \le 3 \cr & \Leftrightarrow - {1 \over {12}} \le k \le {{41} \over {12}} \cr & Ma\,\,k \in Z \Rightarrow k \in \left\{ {0;1;2;3} \right\} \cr & \Rightarrow x \in \left\{ {{{ - 5\pi } \over {12}};{{7\pi } \over {12}};{{19\pi } \over {12}};{{31\pi } \over {12}}} \right\} \cr & \Rightarrow Tong\,\,cac\,\,nghiem\,\,la: \cr & {{5\pi } \over {12}} + {{17\pi } \over {12}} + {{29\pi } \over {12}} + {{ - 5\pi } \over {12}} + {{7\pi } \over {12}} + {{19\pi } \over {12}} + {{31\pi } \over {12}} = {{103\pi } \over {12}} \cr} \)