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\chapHead{Angles orientés}

\section{Déterminer la mesure principale d'un angle}

\paragraph*{Exemple 1.} On pose $\alpha=\dfrac{459\pi}{5}$. On cherche sa mesure principale.

\begin{enumerate}
\item D'abord, on effectue à la calculatrice la division : $459\div5=91,8$.

\item Comme la partie entière du résultat (ici : 91) est \textbf{impaire}, on arrondit à 92. On écrit alors :
\[
\frac{459\pi}{5}=\frac{(92\times5-1)\pi}{5}=\frac{92\times5\pi}{5}-\frac{\pi}{5}=92-\frac{\pi}{5}=46\times2\pi-\frac{\pi}{5}.
\]
\medskip 

La mesure principale de $\alpha$ est donc : $\alpha=-\dfrac{\pi}{5}$.
\end{enumerate}

\paragraph*{Exemple 2.} On pose $\beta=\dfrac{
887\pi}{7}$. On cherche la mesure principale de $\beta$.

\begin{enumerate}
\item D'abord, on effectue à la calculatrice la division : $887\div7=126,7$.

\item Comme la partie entière du résultat (ici : 126) est \textbf{paire}, on arrondit à 126. On écrit alors :
\[
\frac{887\pi}{7}=\frac{(126\times7+5)\pi}{7}=\frac{126\times7\pi}{7}+\frac{5\pi}{7}=126+\frac{5\pi}{7}=63\times2\pi+\frac{5\pi}{7}.
\]
\medskip 

La mesure principale de $\beta$ est donc : $\beta=\dfrac{5\pi}{7}$.
\end{enumerate}

\newpage

\section{Le cercle trigonométrique}

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\bigskip

\arrayrulecolor{violet}
\begin{tabular}{|>{\columncolor{purple!10}}Sr|*5{Sc|}}
\hline
Angles (en radians) & 0 & $\dfrac{\pi}{6}$ & $\dfrac{\pi}{4}$ & $\dfrac{\pi}{3}$ & $\dfrac{\pi}{2}$\\
\hline
Cosinus & 1 & $\dfrac{\sqrt{3}}{2}$ & $\dfrac{\sqrt{2}}{2}$ & $\dfrac{1}{2}$ & 0\\
\hline
Sinus & 0 & $\dfrac{1}{2}$ & $\dfrac{\sqrt{2}}{2}$ & $\dfrac{\sqrt{3}}{2}$ & 1\\
\hline
\end{tabular}
\end{center}

\paragraph*{Relation fondamentale de trigonométrie :} $\cos^2\alpha+\sin^2\alpha=1$ pour tout angle $\alpha$.
\end{document}