a) cos\(\left(\dfrac{2x}{3}+\dfrac{\pi}{4}\right)\)=\(-\dfrac{1}{2}\)
\(\Rightarrow cos\left(\dfrac{2x}{3}+\dfrac{\pi}{4}\right)=cos\dfrac{2\pi}{3}\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{2x}{3}+\dfrac{\pi}{4}=\dfrac{2\pi}{3}+k2\pi\\\dfrac{2x}{3}+\dfrac{\pi}{4}=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{2x}{3}=\dfrac{5\pi}{12}+k2\pi\\\dfrac{2x}{3}=-\dfrac{11\pi}{12}+k2\pi\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{5\pi}{8}+k3\pi\\x=-\dfrac{11\pi}{8}+k3\pi\end{matrix}\right.\)(\(k\in Z)\)
a.
\(\Leftrightarrow cos\left(\dfrac{x}{2}+\dfrac{\pi}{4}\right)=cos\left(\dfrac{2\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}+\dfrac{\pi}{4}=\dfrac{2\pi}{3}+k2\pi\\\dfrac{x}{2}+\dfrac{\pi}{4}=-\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{6}+k4\pi\\x=-\dfrac{11\pi}{6}+k4\pi\end{matrix}\right.\)
b.
\(\Leftrightarrow tan\left(\dfrac{3x}{2}-10^0\right)=tan\left(-30^0\right)\)
\(\Leftrightarrow\dfrac{3x}{2}-10^0=-30^0+k180^0\)
\(\Leftrightarrow\dfrac{3x}{2}=-20^0+k180^0\)
\(\Leftrightarrow x=-\dfrac{40^0}{3}+k120^0\)
\(tan\left(\dfrac{3x}{2}-10^o\right)=-\dfrac{1}{\sqrt{3}}=tan\left(-30^o\right)\)
\(\Rightarrow\dfrac{3x}{2}-10^o=-30^o+k180^o\)
\(\Rightarrow\dfrac{3x}{2}=-20^o+k180^o\)
\(\Rightarrow x=-\dfrac{1}{3}\cdot40^o+k120^o\) \(\left(k\in Z\right)\)