Devoir Surveillé 1 S02 Calcul Trigonométrique A - B
📅 April 20, 2024 | 👁️ Views: 439
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\noindent\begin{tabularx}{\textwidth}{@{} lCr @{}}
Lycee Taghzirt\textbf{/}Prof MOSAID &
2022-2023\textbf{/Devoir 1 S02}\ccc{A}&
TCSF \textbf{/}2h\\
\bottomrule
\end{tabularx}
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\exe{1}(4pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.98\textwidth}}
4& \mylabel[green]{1} Résoudre dans \(\mathbb{R}:\) \(\quad x^2+6x-7 \le 0\quad \) ; \(\quad x^2-3x+2=0\quad\) et dans \(\quad \mathbb{R}^2: \quad \begin{cases}
2x+3y=3\\
x+2y=1
\end{cases}\)
\end{tabular}
\\
\\
\exe{2} (6.5pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.96\textwidth}}
2& \mylabel[green]{1} Placer le point \(A\left(\frac{27\pi}{4}\right)\) sur le cercle trigonométrique \\
1.5& \mylabel[green]{2} Sachant que \(\frac{3\pi}{2}<x<2\pi\) et \(\cos x=\frac{3}{7}\) Calculer \(\sin x\)\\
1.5& \mylabel[green]{3} Calculer \hspace*{0.5cm} \(\cos \frac{87\pi}{4}\)\\
1.5& \mylabel[green]{4} Calculer: \hspace*{0.5cm} \(A = \cos \frac{4\pi}{19} +\cos \frac{5\pi}{19} + \cos \frac{14\pi}{19} + \cos \frac{15\pi}{19} \)
\end{tabular}
\\
\exe{3}(9.5pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.96\textwidth}}
2& \mylabel[green]{1} Résoudre dans \(\mathbb{R}\): \( \tan\left(3x- \frac{\pi}{4}\right)+\sqrt3=0\)\\
3& \mylabel[green]{2.a} Résoudre \(x\in ]-\pi,\pi]\quad 2\cos x -1=0 \qquad\) et \(\qquad x\in ]-\pi,\pi]\quad \sqrt2\sin x -1=0\)\\
3& \mylabel[green]{2.b} Résoudre \(x\in ]-\pi,\pi]\quad 2\cos x -1\ge0 \qquad\) et \(\qquad x\in ]-\pi,\pi]\quad \sqrt2\sin x -1\ge0\)\\
1.5& \mylabel[green]{2.c} En déduir les solutions de l'inéquation: \(\qquad x\in ]-\pi,\pi]\quad \frac{2\cos x -1}{\sqrt2\sin x -1} \ge0\)\\
%3&\mylabel[green]{3} Résoudre \(\qquad x\in ]-\pi,\pi]\quad 2\sin^2(7\pi+x)-3\sqrt3\cos\left(\frac{9\pi}{2}-x\right)+3=0\)
\end{tabular}
\mylink \hfill \mylink\\
\\[2cm]
%==================================================
%==================================================
\noindent\begin{tabularx}{\textwidth}{@{} lCr @{}}
Lycee Taghzirt\textbf{/}Prof MOSAID &
2022-2023\textbf{/Devoir 1 S02}\ccc{B}&
TCSF \textbf{/}2h\\
\bottomrule
\end{tabularx}
\mylink \hfill \mylink\\
\exe{1}(4pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.98\textwidth}}
4& \mylabel[green]{1} Résoudre dans \(\mathbb{R}:\) \(\quad 3x^2-2x-8 \le 0 \) ; \(\quad -x^2+3x-2=0\quad\) et dans \(\quad \mathbb{R}^2: \begin{cases}
2x-3y=1\\
4x+5y=-2
\end{cases}\)
\end{tabular}
\exe{2} (6.5pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.96\textwidth}}
2& \mylabel[green]{1} Placer le point \(A\left(\frac{47\pi}{3}\right)\) sur le cercle trigonométrique \\
1.5& \mylabel[green]{2} Sachant que \(\frac{\pi}{2}<x<\pi\) et \(\sin x=\frac{3}{7}\) Calculer \(\cos x\)\\
1.5& \mylabel[green]{3} Calculer \hspace*{0.5cm} \(\sin \frac{95\pi}{3}\)\\
1.5& \mylabel[green]{4} Simplifier: \hspace*{0.5cm} \(A = \cos\left(x-\frac{97\pi}{2}\right) +\sin\left(x+\frac{95\pi}{2}\right) +\cos(103\pi-x) + \sin(203\pi-x) \)
\end{tabular}
\\
\exe{3}(9.5pts)\\
\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.96\textwidth}}
2& \mylabel[green]{1} Résoudre: \(x\in[-\pi,3\pi] \qquad 2\sin\left(x\right)+\sqrt2=0\)\\
%3& \mylabel[green]{2} Résoudre \(x\in ]-\pi,\pi]\quad 4\sin^2 x +2\left(1-\sqrt2\right)\sin x -\sqrt2=0\)\\
3&\mylabel[green]{2.a} Résoudre \(\quad x\in ]-\pi,2\pi]\quad 2\cos x +\sqrt3\le0\quad\) et \(\quad x\in ]-\pi,2\pi]\quad \sin x \ge \frac{1}{2}\)\\
3& \mylabel[green]{2.b} Résoudre \(\quad x\in ]-\pi,2\pi] \qquad \left(2\cos x+\sqrt3\right)\left(\sin x -\frac{1}{2}\right)>0\)\\
1.5& \mylabel[green]{4} Soit \(x \in \mathbb{R}\) tel que \(x\ne \frac{\pi}{2}+k\pi\) et \(k\in\mathbb{Z}\)\\
& \hspace*{1cm}Montrer que \(\frac{1}{1-\sin x}+ \frac{1}{1-\sin x}-2\tan^2x=2\)
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