Câu hỏi:
Phương trình \({\rm{cos}}2x + \sin x = \sqrt 3 \left( {\cos x – \sin 2x} \right)\) có các nghiệm là:
-
A.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{18}} + k\dfrac{{2\pi }}{3}\\x = \dfrac{{\pi }}{2} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\). -
B.
\(\left[ \begin{array}{l}x = – \dfrac{\pi }{4} + k\pi \\x = – \dfrac{\pi }{{12}} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\). -
C.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{12}} + k\pi \\x = \dfrac{\pi }{4} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\). -
D.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{12}} + k2\pi \\x = – \dfrac{\pi }{4} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\).
Lời giải tham khảo:
Đáp án đúng: A
Ta có: \({\rm{cos}}2x + \sin x = \sqrt 3 \left( {\cos x – \sin 2x} \right)\)
\(\begin{array}{l}
\Leftrightarrow \cos 2x + \sin x = \sqrt 3 \cos x – \sqrt 3 \sin 2x\\
\Leftrightarrow \cos 2x + \sqrt 3 \sin 2x = \sqrt 3 \cos x – \sin x\\
\Leftrightarrow \frac{1}{2}\cos 2x + \frac{{\sqrt 3 }}{2}\sin 2x = \frac{{\sqrt 3 }}{2}\cos x – \frac{1}{2}\sin x
\end{array}\)
\( \Leftrightarrow \cos \left( {2x – \dfrac{\pi }{3}} \right) = \cos \left( {x + \dfrac{\pi }{6}} \right)\)
\( \Leftrightarrow \left[ \begin{array}{l}2x – \dfrac{\pi }{3} = x + \dfrac{\pi }{6} + k2\pi \\2x – \dfrac{\pi }{3} = – x – \dfrac{\pi }{6} + k2\pi \end{array} \right. \) \(\Leftrightarrow \left[ \begin{array}{l}x = \dfrac{\pi }{2} + k2\pi \\x = \dfrac{\pi }{{18}} + k\dfrac{{2\pi }}{3}\end{array} \right.\;\left( {k \in \mathbb{Z}} \right)\)
Chọn đáp án A.
Trả lời