Giải phương trình: sin ( x + pi /3) - cos ( 3x + pi /6) = 0

Trả lời

Lời giải

\(\sin \left( {x + \frac{\pi }{3}} \right) - \cos \left( {3x + \frac{\pi }{6}} \right) = 0\)

\( \Leftrightarrow \sin \left( {x + \frac{\pi }{3}} \right) = \cos \left( {3x + \frac{\pi }{6}} \right)\)

\( \Leftrightarrow \cos \left( {\frac{\pi }{2} - x - \frac{\pi }{3}} \right) = \cos \left( {3x + \frac{\pi }{6}} \right)\)

\( \Leftrightarrow \cos \left( {x - \frac{\pi }{6}} \right) = \cos \left( {3x + \frac{\pi }{6}} \right)\)

\( \Leftrightarrow \left[ \begin{array}{l}3x + \frac{\pi }{6} = x - \frac{\pi }{6} + k2\pi \\3x + \frac{\pi }{6} = - x + \frac{\pi }{6} + k2\pi \end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}2x = - \frac{\pi }{3} + k2\pi \\4x = k2\pi \end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}x = - \frac{\pi }{6} + k\pi \\x = k\frac{\pi }{2}\end{array} \right.\;\left( {k \in \mathbb{Z}} \right)\)