Question

GPT

a) \(2sinx=1\) với \(x\in\left(0;\pi\right)\)

b) \(sin\left(cosx\right)=0\) với \(x\in\left[0;2\pi\right]\)

Answer 1

a/

\(sinx=\frac{1}{2}\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)

\(x\in\left(0;\pi\right)\Rightarrow\left[{}\begin{matrix}0< \frac{\pi}{6}+k2\pi< \pi\\0< \frac{5\pi}{6}+k2\pi< \pi\end{matrix}\right.\) \(\Rightarrow k=0\)

\(\Rightarrow x=\left\{\frac{\pi}{6};\frac{5\pi}{6}\right\}\)

b.

\(\Leftrightarrow cosx=k\pi\)

Do \(-1\le cosx\le1\Rightarrow-1\le k\pi\le1\Rightarrow k=0\)

\(\Rightarrow cosx=0\Rightarrow x=\frac{\pi}{2}+k\pi\)

\(\Rightarrow x=\left\{\frac{\pi}{2};\frac{3\pi}{2}\right\}\)