Giải phương trình: sin 2x + cos x + cos 2x - sin x = 0

Trả lời

Lời giải

Ta có \(\sin 2x + \cos x + \cos 2x - \sin x = 0\)

Û sin 2x + cos 2x = sin x − cos x

\[ \Leftrightarrow \sqrt 2 \sin \left( {2x + \frac{\pi }{4}} \right) = \sqrt 2 \sin \left( {x - \frac{\pi }{4}} \right)\]

\[ \Leftrightarrow \sin \left( {2x + \frac{\pi }{4}} \right) = \sin \left( {x - \frac{\pi }{4}} \right)\]

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

\( \Leftrightarrow \left[ \begin{array}{l}x = - \frac{\pi }{2} + k2\pi \\x = \frac{\pi }{3} + \frac{{k2\pi }}{3}\end{array} \right.\;\left( {k \in \mathbb{Z}} \right)\).