Đáp án:

\(\eqalign{ & a)\,\,x = {\pi \over 6} + k2\pi \,\,\left( {k \in Z} \right) \cr & b)\,\,\left[ \matrix{ x = {{5\pi } \over {84}} + {{k2\pi } \over 7} \hfill \cr x = - {{13\pi } \over {84}} + {{k2\pi } \over 7} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr & c)\,\,\left[ \matrix{ x = {\pi \over 3} + k\pi \hfill \cr x = {\pi \over 3} + {{k\pi } \over 2} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} \)

Giải thích các bước giải:

$$\eqalign{ & a)\,\,\sin x + \sqrt 3 \cos x = 2 \cr & \Leftrightarrow {1 \over 2}\sin x + {{\sqrt 3 } \over 2}\cos x = 1 \cr & \Leftrightarrow \sin x\cos {\pi \over 3} + \cos x\sin {\pi \over 3} = 1 \cr & \Leftrightarrow \sin \left( {x + {\pi \over 3}} \right) = 1 \cr & \Leftrightarrow x + {\pi \over 3} = {\pi \over 2} + k2\pi \cr & \Leftrightarrow x = {\pi \over 6} + k2\pi \,\,\left( {k \in Z} \right) \cr & b)\,\,\cos 7x - \sqrt 3 \sin 7x = - \sqrt 2 \cr & \Leftrightarrow {1 \over 2}\cos 7x - {{\sqrt 3 } \over 2}\sin 7x = - {{\sqrt 2 } \over 2} \cr & \Leftrightarrow \cos 7x\cos {\pi \over 3} - \sin 7x\sin {\pi \over 3} = - {{\sqrt 2 } \over 2} \cr & \Leftrightarrow \cos \left( {7x + {\pi \over 3}} \right) = - {{\sqrt 2 } \over 2} \cr & \Leftrightarrow \left[ \matrix{ 7x + {\pi \over 3} = {{3\pi } \over 4} + k2\pi \hfill \cr 7x + {\pi \over 3} = - {{3\pi } \over 4} + k2\pi \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{ 7x = {{5\pi } \over {12}} + k2\pi \hfill \cr 7x = - {{13\pi } \over {12}} + k2\pi \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{ x = {{5\pi } \over {84}} + {{k2\pi } \over 7} \hfill \cr x = - {{13\pi } \over {84}} + {{k2\pi } \over 7} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr & c)\,\,2\cos 3x + \sqrt 3 \sin x + \cos x = 0 \cr & \Leftrightarrow {{\sqrt 3 } \over 2}\sin x + {1 \over 2}\cos x = - \cos 3x \cr & \Leftrightarrow \sin x\cos {\pi \over 6} + \cos x\sin {\pi \over 6} = - \cos 3x \cr & \Leftrightarrow \sin \left( {x + {\pi \over 6}} \right) = \sin \left( {3x - {\pi \over 2}} \right) \cr & \Leftrightarrow \left[ \matrix{ x + {\pi \over 6} = 3x - {\pi \over 2} + k2\pi \hfill \cr x + {\pi \over 6} = \pi - 3x + {\pi \over 2} + k2\pi \hfill \cr} \right. \cr & \Leftrightarrow \left[ \matrix{ 2x = {{2\pi } \over 3} + k2\pi \hfill \cr 4x = {{4\pi } \over 3} + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{ x = {\pi \over 3} + k\pi \hfill \cr x = {\pi \over 3} + {{k\pi } \over 2} \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr} $$