1.
\(sin3x=-\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}3x=-\dfrac{\pi}{6}+k2\pi\\3x=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{18}+\dfrac{k2\pi}{3}\\x=\dfrac{7\pi}{18}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
2.
\(cos2x=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow2x=\pm\dfrac{3\pi}{4}+k2\pi\)
\(\Leftrightarrow x=\pm\dfrac{3\pi}{8}+k\pi\)
4.
\(cos\left(2x+\dfrac{\pi}{6}\right)=\dfrac{1}{3}\)
\(\Leftrightarrow2x+\dfrac{\pi}{6}=\pm arccos\dfrac{1}{3}+k2\pi\)
\(\Leftrightarrow x=-\dfrac{\pi}{12}\pm arccos\dfrac{1}{3}+k\pi\)
6.
\(tan\left(x-\dfrac{\pi}{4}\right)=\sqrt{3}\)
\(\Leftrightarrow x-\dfrac{\pi}{4}=\dfrac{\pi}{3}+k\pi\)
\(\Leftrightarrow x=\dfrac{7\pi}{12}+k\pi\)
7.
\(tan\left(\dfrac{\pi}{5}-x\right)=2\)
\(\Leftrightarrow\dfrac{\pi}{5}-x=arctan2+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{5}-arctan2+k\pi\)
8.
\(cot\left(\dfrac{\pi}{4}-x\right)=\dfrac{1}{\sqrt{3}}\)
\(\Leftrightarrow\dfrac{\pi}{4}-x=\dfrac{\pi}{3}+k\pi\)
\(\Leftrightarrow x=-\dfrac{\pi}{12}+k\pi\)
9.
\(cos\dfrac{x}{2}=cos\sqrt{2}\)
\(\Leftrightarrow\dfrac{x}{2}=\pm\sqrt{2}+k2\pi\)
\(\Leftrightarrow x=\pm2\sqrt{2}+k2\pi\)
10.
\(sin\dfrac{x+\pi}{5}=-\dfrac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x+\pi}{5}=-\dfrac{\pi}{6}+k2\pi\\\dfrac{x+\pi}{5}=\dfrac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{11\pi}{6}+k2\pi\\x=\dfrac{29\pi}{6}+k2\pi\end{matrix}\right.\)
11.
\(sin\left(2x-\dfrac{\pi}{5}\right)=sin\left(\dfrac{\pi}{5}+x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{5}=\dfrac{\pi}{5}+x+k2\pi\\2x-\dfrac{\pi}{5}=\pi-\dfrac{\pi}{5}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{2\pi}{5}+k2\pi\\x=\dfrac{\pi}{3}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
