Số nghiệm của phương trình \(\sin 5x + \sqrt 3 \co...

A

- Hướng dẫn giải

Ta có: \(\sin 5x + \sqrt 3 \cos 5x = 2\sin 7x \Leftrightarrow \sin \left( {5x + \frac{\pi }{3}} \right) = \sin 7x\) 

\( \Leftrightarrow \left[ \begin{array}{l}
7x = 5x + \frac{\pi }{3} + k2\pi \\
7x = \pi  - 5x - \frac{\pi }{3} + k2\pi 
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{6} + k\pi \\
x = \frac{\pi }{{18}} + k\frac{\pi }{6}
\end{array} \right.,k \in Z\) 

TH1: \(0 < \frac{\pi }{6} + k\pi  < \frac{\pi }{2} \Leftrightarrow  - \frac{1}{6} < k < \frac{1}{3} \Rightarrow k = 0 \Rightarrow x = \frac{\pi }{6}\) 

TH2: \(0 < \frac{\pi }{{18}} + k\frac{\pi }{6} < \frac{\pi }{2} \Leftrightarrow 0 < \frac{1}{3} + k < 3 \Leftrightarrow  - \frac{1}{3} < k < 3 - \frac{1}{3} \Rightarrow k = 0,1,2 \Rightarrow x = \frac{\pi }{{18}},\frac{{2\pi }}{9},\frac{{7\pi }}{{18}}.\) 

Vậy \(x \in \left\{ {\frac{\pi }{{18}},\frac{{2\pi }}{9},\frac{{7\pi }}{{18}},\frac{\pi }{6}} \right\}.\)