Phương trình \[\sin 2x = \frac{1}{2}\] có bao nhiêu nghiệm trên khoảng \[\left( {0;\frac{{15\pi }}{2}} \right)\]?

Chọn B

\[\sin 2x = \frac{1}{2} \Leftrightarrow \sin 2x = \sin \frac{\pi }{6} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{2x = \frac{\pi }{6} + k2\pi }\\{2x = \pi - \frac{\pi }{6} + k2\pi }\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{{12}} + k\pi }\\{x = \frac{{5\pi }}{{12}} + k\pi }\end{array}} \right.\left( {k \in Z} \right)\]

Vì \[x \in \left( {0;\frac{{15\pi }}{2}} \right) \Rightarrow 0 < \frac{\pi }{{12}} + k\pi < \frac{{15\pi }}{2} \Leftrightarrow \frac{{ - 1}}{{12}} < k < \frac{{89}}{{12}} \Rightarrow k \in \left\{ {0;1;2;3;4;5;6;7} \right\}\]

\(0 < \)\[0 < \frac{{5\pi }}{{12}} + k\pi < \frac{{15\pi }}{2} \Leftrightarrow \frac{{ - 5}}{{12}} < k < \frac{{15}}{2} \Rightarrow k \in \left\{ {0;1;2;3;4;5;6;7} \right\}\].