Giải phương trình lượng giác \(\sin \left( {\dfrac{\pi }{3} - 3x} \right) = \sin \left( {x + \dfrac{\pi }{4}} \right)\) có nghiệm là:
-
A.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{48}} + \dfrac{{k\pi }}{2}\\x = - \dfrac{{5\pi }}{{24}} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\)
-
B.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{48}} + k\pi \\x = - \dfrac{{5\pi }}{{12}} + k2\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\)
-
C.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{24}} + k\pi \\x = - \dfrac{{5\pi }}{{48}} + \dfrac{{k\pi }}{2}\end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\)
-
D.
\(\left[ \begin{array}{l}x = \dfrac{\pi }{{24}} + k2\pi \\x = - \dfrac{{5\pi }}{{48}} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\)
Giải phương trình lượng giác cơ bản: \(\sin x = \sin \alpha \Leftrightarrow \left[ \begin{array}{l}x = \alpha + k2\pi \\x = \pi - \alpha + k2\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\).
\(\begin{array}{l}\sin \left( {\dfrac{\pi }{3} - 3x} \right) = \sin \left( {x + \dfrac{\pi }{4}} \right) \Leftrightarrow \left[ \begin{array}{l}\dfrac{\pi }{3} - 3x = x + \dfrac{\pi }{4} + k2\pi \\\dfrac{\pi }{3} - 3x = \pi - x - \dfrac{\pi }{4} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} - 4x = - \dfrac{\pi }{{12}} + k2\pi \\ - 2x = \dfrac{{5\pi }}{{12}} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \dfrac{\pi }{{48}} + \dfrac{{k\pi }}{2}\\x = - \dfrac{{5\pi }}{{24}} + k\pi \end{array} \right.\,\,\left( {k \in \mathbb{Z}} \right)\end{array}\)
Đáp án : A
