$\begin{array}{l}
a)\sin \left( {x + \frac{\pi }{4}} \right) = \sin \left( { - 2x + \frac{\pi }{3}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
x + \frac{\pi }{4} = - 2x + \frac{\pi }{3} + k2\pi \\
x + \frac{\pi }{4} = \pi + 2x - \frac{\pi }{3} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
3x = \frac{\pi }{{12}} + k2\pi \\
- x = \frac{{5\pi }}{{12}} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{{36}} + \frac{{k2\pi }}{3}\\
x = - \frac{{5\pi }}{{12}} - k2\pi
\end{array} \right.\\
b)\sin \left( {2x - {{60}^0}} \right) = \frac{{\sqrt 3 }}{2} = \sin {60^0}\\
\Leftrightarrow \left[ \begin{array}{l}
2x - {60^0} = {60^0} + k{360^0}\\
2x - {60^0} = {180^0} - {60^0} + k{360^0}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = {120^0} + k{360^0}\\
2x = {180^0} + k{360^0}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = {60^0} + k{180^0}\\
x = {90^0} + k{180^0}
\end{array} \right.\\
c)\cos 3x = - \frac{1}{2} = \cos \frac{{2\pi }}{3}\\
\Leftrightarrow 3x = \pm \frac{{2\pi }}{3} + k2\pi \\
\Leftrightarrow x = \pm \frac{{2\pi }}{9} + \frac{{k2\pi }}{3}\\
d)\cos \left( {x + {{20}^0}} \right) = \cos \left( {{{30}^0} - x} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
x + {20^0} = {30^0} - x + k{360^0}\\
x + {20^0} = - {30^0} + x + k{360^0}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = {10^0} + k{360^0}\\
0 = - {50^0} + k{360^0}\left( {VN} \right)
\end{array} \right. \Leftrightarrow x = {5^0} + k{180^0}\\
e)\tan 2x = \frac{1}{2}\\
\Leftrightarrow 2x = \arctan \frac{1}{2} + k\pi \\
\Leftrightarrow x = \frac{1}{2}\arctan \frac{1}{2} + \frac{{k\pi }}{2}\\
f)\cot \left( {x + 1} \right) = - \frac{{\sqrt 3 }}{3} = \cot \left( { - \frac{\pi }{3}} \right)\\
\Leftrightarrow x + 1 = - \frac{\pi }{3} + k\pi \\
\Leftrightarrow x = - \frac{\pi }{3} - 1 + k\pi
\end{array}$
