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\begin{document}


\begin{claim}
\begin{enumerate}
\item \label{un_echg} 
\begin{equation*}
\cos x=\sqrt{3}/2 \Leftrightarrow x=\pm \frac{\pi}{6}\mod{2\pi}
\Leftrightarrow x=\pm\frac{\pi}{6}\text{ ou }-\frac{\pi}{6}-2\pi=-\frac{11\pi%
}{6}\text{ sur }[-2\pi;\pi[.
\end{equation*}
D'où $\mathcal{S}=\left\{\dfrac{\pi}{6}\,;\,-\dfrac{\pi}{6}\,;\,-\dfrac{11\pi%
}{6}\right\}$.

\item \label{deux_echg} 
\begin{equation*}
\sin x=\frac{1}{2} \Leftrightarrow x=\frac{\pi}{6}\mod{2\pi} \text{ ou } \pi-%
\frac{\pi}{6}=\frac{5\pi}{6}\mod{2\pi} \Leftrightarrow x=\frac{\pi}{6}\text{
ou }\frac{-11\pi}{6}\text{ ou }\frac{5\pi}{6}\text{ ou }-\frac{7\pi}{6}\text{
sur }[-2\pi;\pi[.
\end{equation*}
D'où $\mathcal{S}=\left\{\dfrac{\pi}{6}\,;\,-\dfrac{11\pi}{6}\,;\,\dfrac{5\pi%
}{6}\,;\,-\frac{7\pi}{6}\right\}$.

\item \quad 
\begin{minipage}[c]{8cm}
  \begin{eqnarray*}
    &&\sin x>1/2\\
    &\Leftrightarrow& x\in
    \left]\frac{\pi}{6}+k2\pi\,;\,\frac{5\pi}{6}+k2\pi\right[\text{ où
    }k\in\mathbb{Z}\\
    &\Leftrightarrow&x\in\left]-\frac{11\pi}{6}\,;\,-\frac{7\pi}{6}\right[\cup\left]\frac{\pi}{6}\,;\,\frac{5\pi}{6}\right[ \text{ sur } \left[-2\pi\,;\,\pi\right[
  \end{eqnarray*}
  D'où $\mathcal{S}=\left]-\frac{11\pi}{6}\,;\,-\frac{7\pi}{6}\right[\cup\left]\frac{\pi}{6}\,;\,\frac{5\pi}{6}\right[$.
  \end{minipage}\hfill 
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\item 
\begin{eqnarray*}
&&\cos x\leq \sqrt{3}/2 \\
&\Leftrightarrow &x\in \left] \frac{\pi }{6}+k2\pi \,;\,\frac{11\pi }{6}%
+k2\pi \right[ \text{ où }k\in \mathbb{Z} \\
&\Leftrightarrow &x\in \left] \begin{minipage}[c]{0cm}
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\setlength{\fboxsep}{0pt} \fbox{\rule{0pt}{10mm}} \end{minipage}%
\makebox[1cm][l]{\dotfill }\;\,;\,\makebox[1cm][l]{\dotfill }\;\right[ \text{
sur }\left[ -2\pi \,;\,\pi \right[ 
\end{eqnarray*}

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Figures à compléter

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\vspace{1cm} D'où $\mathcal{S}=\makebox[6cm][l]{\dotfill }\;$.

\item 
\begin{equation*}
\cos x=-\sqrt{3}/2\Leftrightarrow x=\pm \frac{5\pi }{6}\quad \lbrack 2\pi
]\Leftrightarrow x=-\frac{5\pi }{6}\text{ ou }\frac{5\pi }{6}\text{ ou }-%
\frac{7\pi }{6}\text{ sur }[-2\pi ;\pi \lbrack .
\end{equation*}%
D'où $\mathcal{S}=\left\{ -\frac{5\pi }{6}\,;\,\frac{5\pi }{6}\,;\,-\frac{%
7\pi }{6}\right\} $.

\item \label{six_echg} 
\begin{eqnarray*}
\sin x=-1/2 &\Leftrightarrow &x=-\frac{\pi }{6}\quad \lbrack 2\pi ]\text{ ou 
}\pi +\frac{\pi }{6}\quad \lbrack 2\pi ]\Leftrightarrow x=-\frac{\pi }{6}%
\quad \lbrack 2\pi ]\text{ ou }\frac{7\pi }{6}\quad \lbrack 2\pi ] \\
&\Leftrightarrow &x=-\frac{\pi }{6}\text{ ou }-\frac{5\pi }{6}\text{ sur }%
[-2\pi ;\pi \lbrack .
\end{eqnarray*}%
D'où $\mathcal{S}=\left\{ -\frac{\pi }{6}\,;\,-\frac{5\pi }{6}\right\} $.

\item D'après le \ref{un_echg}) et le \ref{six_echg}) on a : 
\begin{equation*}
\mathcal{S}=\left\{ -\frac{\pi }{6}\right\} .
\end{equation*}

\item D'après le \ref{un_echg}) et le \ref{deux_echg}) on a : 
\begin{equation*}
\mathcal{S}=\left\{ \frac{\pi }{6}\,;\,-\frac{11\pi }{6}\right\} .
\end{equation*}
\end{enumerate}
\end{claim}

\end{document}
