\begin{enumerate}
\item $\begin{aligned}[t]
\cos\left(2x+\dfrac{\pi}{6}\right)=\dfrac{1}{2}  
& \iff \cos\left(2x+\dfrac{\pi}{6}\right)=\cos\left(\dfrac{\pi}{3}\right)\\
& \iff 
\begin{cases} 
2x+\dfrac{\pi}{6}=\dfrac{\pi}{3}+2k\pi\\[8pt]
2x+\dfrac{\pi}{6}=-\dfrac{\pi}{3}+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
2x=\dfrac{\pi}{3}-\dfrac{\pi}{6}+2k\pi\\[8pt]
2x=-\dfrac{\pi}{3}-\dfrac{\pi}{6}+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
2x=\dfrac{\pi}{6}+2k\pi\\[8pt]
2x=-\dfrac{\pi}{2}+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
x=\dfrac{\pi}{12}+k\pi\\[8pt]
x=-\dfrac{\pi}{4}+k\pi
\end{cases}
\end{aligned}$

\begin{minipage}{7cm}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\draw[->,>=latex] (-3,0) -- (3,0) node[below left] {cos};
\draw[->,>=latex] (0,-3) -- (0,3) node[right] {sin};
\node[below left] at (0,0) {O};
\fill[purple] ({180/12}:2cm) circle (1.5pt) node[right] {$\frac{\pi}{12}$};
\fill[purple] ({180/12+180}:2cm) circle (1.5pt) node[left] {$-\frac{11\pi}{12}$};
\fill[violet] (-45:2cm) circle (1.5pt) node[right] {$-\frac{\pi}{4}$};
\fill[violet] ({-45+180}:2cm) circle (1.5pt) node[left] {$\frac{3\pi}{4}$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{\dimexpr\linewidth-7.2cm}
Sur \intervOF{-\pi}{\pi}, les solutions sont donc :
\begin{itemize}
\item $\dfrac{\pi}{12}$
\item $\dfrac{\pi}{12}-\pi=-\dfrac{11\pi}{12}$
\item $-\dfrac{\pi}{4}$
\item $-\dfrac{\pi}{4}+\pi=\dfrac{3\pi}{4}$
\end{itemize}

$\boxed{\mathcal{S}=\left\{-\dfrac{11\pi}{12}~;~-\dfrac{\pi}{4}~;~\dfrac{\pi}{12}~;~\dfrac{3\pi}{4}\right\}}$
\end{minipage}
\newpage
\item $\begin{aligned}[t]
\sin\left(3x-\dfrac{\pi}{7}\right)=-\dfrac{\sqrt{2}}{2}
& \iff \sin\left(3x-\dfrac{\pi}{7}\right)=\sin\left(-\dfrac{\pi}{4}\right)\\
& \iff
\begin{cases}
3x-\dfrac{\pi}{7}=-\dfrac{\pi}{4}+2k\pi\\[8pt]
3x-\dfrac{\pi}{7}=\pi-\left(-\dfrac{\pi}{4}\right)+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
3x=-\dfrac{\pi}{4}+\dfrac{\pi}{7}+2k\pi\\[8pt]
3x=\dfrac{5\pi}{4}+\dfrac{\pi}{7}+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
3x=-\dfrac{3\pi}{28}+2k\pi\\[8pt]
3x=\dfrac{39\pi}{28}+2k\pi
\end{cases}
\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
x=-\dfrac{\pi}{28}+\dfrac{2k\pi}{3}\\[8pt]
x=\dfrac{13\pi}{28}+\dfrac{2k\pi}{3}
\end{cases}
\quad k\in\Z
\end{aligned}$

\begin{minipage}{7cm}
\begin{tikzpicture}
\draw (0,0) circle (2cm);
\draw[->,>=latex] (-3,0) -- (3,0) node[below left] {cos};
\draw[->,>=latex] (0,-3) -- (0,3) node[right] {sin};
\node[below left] at (0,0) {O};
\fill[purple] ({-180/28}:2cm) circle (1.5pt) node[below right] {$-\frac{\pi}{28}$};
\fill[purple] ({-180/28+120}:2cm) circle (1.5pt) node[above left] {$\frac{53\pi}{84}$};
\fill[purple] ({-180/28+240}:2cm) circle (1.5pt) node[below left] {$-\frac{59\pi}{84}$};
\fill[violet] ({13*180/28}:2cm) circle (1.5pt) node[above right] {$\frac{13\pi}{28}$};
\fill[violet] ({13*180/28-120}:2cm) circle (1.5pt) node[right] {$-\frac{17\pi}{84}$};
\fill[violet] ({13*180/28-240}:2cm) circle (1.5pt) node[left] {$-\frac{73\pi}{84}$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{\dimexpr\linewidth-7.2cm}
Sur \intervOF{-\pi}{\pi}, les solutions sont donc :
\begin{itemize}
\item $-\dfrac{\pi}{28}$ , $-\dfrac{\pi}{28}+\dfrac{2\pi}{3}=\dfrac{53\pi}{84}$ et
\item[] $-\dfrac{\pi}{28}-\dfrac{2\pi}{3}=-\dfrac{59\pi}{84}$
\item $\dfrac{13\pi}{28}$ , $\dfrac{13\pi}{28}-\dfrac{2\pi}{3}=-\dfrac{17\pi}{84}$ et 
\item[] $\dfrac{13\pi}{28}-\dfrac{4\pi}{3}=-\dfrac{73\pi}{84}$
\end{itemize}
\end{minipage}
\[
\boxed{\mathcal{S}=\left\{-\dfrac{73\pi}{84}~;~-\dfrac{59\pi}{84}~;~-\dfrac{17\pi}{84}~;~-\dfrac{\pi}{28}~;~\dfrac{13\pi}{28}~;~\dfrac{53\pi}{84}\right\}}
\]

\item $\begin{aligned}[t]
\cos\left(x+\dfrac{\pi}{3}\right)=\sin x & \iff \cos\left(x+\dfrac{\pi}{3}\right)=\cos\left(\dfrac{\pi}{2}-x\right)\\
& \iff
\begin{cases}
x+\dfrac{\pi}{3}=\dfrac{\pi}{2}-x+2k\pi\\[8pt]
x+\dfrac{\pi}{3}=-\dfrac{\pi}{2}+x+2k\pi
\end{cases}\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
2x=\dfrac{\pi}{6}+2k\pi\\[8pt]
\dfrac{\pi}{3}=-\dfrac{\pi}{2}+2k\pi \longleftarrow \text{impossible}
\end{cases}\quad k\in\Z\\[10pt]
& \iff x=\dfrac{\pi}{12}+k\pi\quad , \quad k\in\Z
\end{aligned}$

Sur \intervOF{-\pi}{\pi},\quad$\boxed{\mathcal{S}=\left\{-\dfrac{11\pi}{12}~;~\dfrac{\pi}{12} \right\}}$

\item $\begin{aligned}[t]
\cos\left(2x+\dfrac{\pi}{4}\right)=\sin\left(x-\dfrac{\pi}{3}\right) & \iff \cos\left(2x+\dfrac{\pi}{4}\right)=\cos\left(\dfrac{\pi}{2}-x+\dfrac{\pi}{3}\right)\\
& \iff \cos\left(2x+\dfrac{\pi}{4}\right)=\cos\left(\dfrac{5\pi}{6}-x\right)\\
& \iff 
\begin{cases}
2x+\dfrac{\pi}{4}=\dfrac{5\pi}{6}-x+2k\pi\\[8pt]
2x+\dfrac{\pi}{4}=-\dfrac{5\pi}{6}+x+2k\pi
\end{cases}\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
3x=\dfrac{5\pi}{6}-\dfrac{\pi}{4}+2k\pi\\[6pt]
x=-\dfrac{5\pi}{6}-\dfrac{\pi}{4}+2k\pi
\end{cases}\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
3x=\dfrac{7\pi}{12}+2k\pi\\[6pt]
x=-\dfrac{13\pi}{12}+2k\pi
\end{cases}\quad k\in\Z\\[10pt]
& \iff
\begin{cases}
x=\dfrac{7\pi}{36}+\dfrac{2k\pi}{3}\\[6pt]
x=-\dfrac{13\pi}{12}+2k\pi
\end{cases}\quad k\in\Z
\end{aligned}$

Sur \intervOF{-\pi}{\pi}, \quad $\boxed{\mathcal{S}=\left\{\dfrac{11\pi}{12}~;~\dfrac{7\pi}{36}~;~\dfrac{31\pi}{36}~;~-\dfrac{17\pi}{36} \right\}}$
\end{enumerate}