Phương trình cos 2x + sin x = căn 3. (cos x – sin 2x) có các nghiệm là? 04/07/2021 Bởi aihong Phương trình cos 2x + sin x = căn 3. (cos x – sin 2x) có các nghiệm là?
Đáp án: $\left[\begin{array}{l}x =\dfrac{\pi}{2} +k2\pi\\x= \dfrac{\pi}{18} + k\dfrac{2\pi}{3}\end{array}\right.\quad (k \in \Bbb Z)$ Giải thích các bước giải: $\begin{array}{l}\cos2x + \sin x = \sqrt3(\cos x – \sin2x)\\ \Leftrightarrow \cos2x + \sqrt3\sin2x = \sqrt3\cos x – \sin x\\ \Leftrightarrow \dfrac{1}{2}\cos2x + \dfrac{\sqrt3}{2}\sin2x = \dfrac{\sqrt3}{2}\cos x – \dfrac{1}{2}\sin x\\ \Leftrightarrow \cos\left(2x – \dfrac{\pi}{3}\right) = \cos\left(x + \dfrac{\pi}{6}\right)\\ \Leftrightarrow \left[\begin{array}{l}2x – \dfrac{\pi}{3} = x + \dfrac{\pi}{6} +k2\pi\\2x – \dfrac{\pi}{3} = – x – \dfrac{\pi}{6} + k2\pi\end{array}\right.\\ \Leftrightarrow \left[\begin{array}{l}x =\dfrac{\pi}{2} +k2\pi\\x= \dfrac{\pi}{18} + k\dfrac{2\pi}{3}\end{array}\right.\quad (k \in \Bbb Z) \end{array}$ Bình luận
Đáp án:
$\left[\begin{array}{l}x =\dfrac{\pi}{2} +k2\pi\\x= \dfrac{\pi}{18} + k\dfrac{2\pi}{3}\end{array}\right.\quad (k \in \Bbb Z)$
Giải thích các bước giải:
$\begin{array}{l}\cos2x + \sin x = \sqrt3(\cos x – \sin2x)\\ \Leftrightarrow \cos2x + \sqrt3\sin2x = \sqrt3\cos x – \sin x\\ \Leftrightarrow \dfrac{1}{2}\cos2x + \dfrac{\sqrt3}{2}\sin2x = \dfrac{\sqrt3}{2}\cos x – \dfrac{1}{2}\sin x\\ \Leftrightarrow \cos\left(2x – \dfrac{\pi}{3}\right) = \cos\left(x + \dfrac{\pi}{6}\right)\\ \Leftrightarrow \left[\begin{array}{l}2x – \dfrac{\pi}{3} = x + \dfrac{\pi}{6} +k2\pi\\2x – \dfrac{\pi}{3} = – x – \dfrac{\pi}{6} + k2\pi\end{array}\right.\\ \Leftrightarrow \left[\begin{array}{l}x =\dfrac{\pi}{2} +k2\pi\\x= \dfrac{\pi}{18} + k\dfrac{2\pi}{3}\end{array}\right.\quad (k \in \Bbb Z) \end{array}$