Phương trình \(\cos x + \sqrt 3 \sin x = 3 - {3 \o...

B

- Hướng dẫn giải

Phương pháp giải:

Giải chi tiết:

ĐK:

\(\eqalign{& \,\,\,\,\,\,\cos x + \sqrt 3 \sin x + 1 \ne 0 \cr & \Leftrightarrow 2\left( {{{\sqrt 3 } \over 2}\sin x + {1 \over 2}\cos x}\right) \ne - 1 \cr & \Leftrightarrow 2\left( {\sin x\cos {\pi \over 6} + \cos x\sin {\pi \over 6}} \right) \ne - 1 \cr & \, \Leftrightarrow 2\sin \left( {x + {\pi \over 6}} \right) \ne - 1 \cr & \Leftrightarrow \sin \left( {x + {\pi \over 6}} \right) \ne - {1 \over 2} = \sin \left( {{{ - \pi } \over 6}} \right) \cr & \Leftrightarrow \left\{ \matrix{x + {\pi \over 6} \ne {{ - \pi } \over 6} + k2\pi \hfill \cr x + {\pi \over 6} \ne \pi - {{ - \pi } \over 6} + k2\pi \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{x \ne {{ - \pi } \over 3} + k2\pi \hfill \cr x \ne \pi + k2\pi \hfill \cr} \right. \cr} \)

 Đặt  \(t = \cos x + \sqrt 3 \sin x = 2\left( {{{\sqrt 3 } \over 2}\sin x + {1 \over 2}\cos x} \right) = 2\left( {\sin x\cos {\pi  \over 6} + \cos x\sin {\pi  \over 6}} \right) = 2\sin \left( {x + {\pi  \over 6}} \right) \in \left[ { - 2;2} \right]\backslash \left\{ { - 1} \right\}\)

Khi đó phương trình đã cho

\(\eqalign{& \Leftrightarrow t = 3 - {3 \over {t + 1}} \Leftrightarrow {t^2} + t = 3t + 3 - 3 \Leftrightarrow {t^2} - 2t = 0 \Leftrightarrow \left[ \matrix{t = 0\,\,\left( {tm} \right) \hfill \cr t = 2\,\,\,\left( {tm} \right) \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{2\sin \left( {x + {\pi \over 6}} \right) = 0 \hfill \cr 2\sin \left( {x + {\pi \over 6}} \right) = 2 \hfill \cr} \right. \Leftrightarrow \left[ \matrix{\sin \left( {x + {\pi \over 6}} \right) = 0 \hfill \cr \sin \left( {x + {\pi \over 6}} \right) = 1 \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{ x + {\pi \over 6} = k\pi \hfill \cr x + {\pi \over 6} = {\pi \over 2} + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{x = - {\pi \over 6} + k\pi \hfill \cr x = {\pi \over 3} + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{x = {{5\pi } \over 6} + k\pi \hfill \cr x = {\pi \over 3} + k2\pi \hfill \cr} \right.\,\,\,\left( {k \in Z} \right) \cr} \)

Chọn B.