Question

\(\text{Phương trình cos5x=\dfrac{1}{\sqrt{2}} có bao nhiêu nghiệm thuộc đoạn [−50\pi ;0] ?}\)

Answer 1

\(cos5x=\dfrac{1}{\sqrt{2}}\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{20}+\dfrac{k2\pi}{5}\\x=-\dfrac{\pi}{20}+\dfrac{n2\pi}{5}\end{matrix}\right.\)

\(-50\pi\le x\le0\Rightarrow\left\{{}\begin{matrix}-50\pi\le\dfrac{\pi}{20}+\dfrac{k2\pi}{5}\le0\\-50\pi\le-\dfrac{\pi}{20}+n2\pi\le0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}-\dfrac{1001}{8}\le k\le-\dfrac{1}{8}\\-\dfrac{999}{8}\le n\le\dfrac{1}{8}\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}k=\left\{-125;-124;...;-1\right\}\\n=\left\{-124;-123;...;0\right\}\end{matrix}\right.\)

Có 250 nghiệm