Question

Pt \(2sin2x+\sqrt{2}=0\) có bao nhiêu nghiệm thuộc khoảng \(\left(0;\pi\right)\)

Answer 1

\(\Leftrightarrow sin2x=-\frac{\sqrt{2}}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=-\frac{\pi}{4}+k2\pi\\2x=\frac{5\pi}{4}+n2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{8}+k\pi\\x=\frac{5\pi}{8}+n\pi\end{matrix}\right.\)

Do \(0< x< \pi\)\(\Rightarrow\left[{}\begin{matrix}0< -\frac{\pi}{8}+k\pi< \pi\\0< \frac{5\pi}{8}+n\pi< \pi\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}\frac{1}{8}< k< \frac{9}{8}\\-\frac{5}{8}< n< \frac{3}{8}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}k=1\\n=0\end{matrix}\right.\)

\(\Rightarrow\) Có 2 nghiệm \(x=\left\{\frac{7\pi}{8};\frac{5\pi}{8}\right\}\)