Phương trình \(\cos 2\left(x+\frac{\pi}{3}\right)+4 \cos \left(\frac{\pi}{6}-x\right)=\frac{5}{2}\) có nghiệm là:
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Lời giải:
Báo saiTXĐ: \(D=\mathbb{R}\)
Ta có:
\(\cos 2\left(x+\frac{\pi}{3}\right)+4 \cos \left(\frac{\pi}{6}-x\right)=\frac{5}{2} \Leftrightarrow 1-2 \sin ^{2}\left(x+\frac{\pi}{3}\right)+4 \cos \left(\frac{\pi}{2}-\left(x+\frac{\pi}{3}\right)\right)=\frac{5}{2}\)
\(\begin{array}{l} \Leftrightarrow 1-2 \sin ^{2}\left(x+\frac{\pi}{3}\right)+4 \sin \left(x+\frac{\pi}{3}\right)=\frac{5}{2} \Leftrightarrow 2 \sin ^{2}\left(x+\frac{\pi}{3}\right)-4 \sin \left(x+\frac{\pi}{3}\right)+\frac{3}{2}=0 \\ \Leftrightarrow\left[\begin{array}{c} \sin \left(x+\frac{\pi}{3}\right)=\frac{3}{2} \\ \sin \left(x+\frac{\pi}{3}\right)=\frac{1}{2} \end{array} \Leftrightarrow \sin \left(x+\frac{\pi}{3}\right)=\sin \frac{\pi}{6} \Leftrightarrow\left[\begin{array}{c} x+\frac{\pi}{3}=\frac{\pi}{6}+k 2 \pi \\ x+\frac{\pi}{3}=\frac{5 \pi}{6}+k 2 \pi \end{array} \Leftrightarrow\left[\begin{array}{c} x=-\frac{\pi}{6}+k 2 \pi \\ x=\frac{\pi}{2}+k 2 \pi \end{array},(k \in \mathbb{Z})\right.\right.\right. \end{array}\)
