Devoir Surveillé 1 S02 Calcul Trigonométrique C - D
📅 April 23, 2024 | 👁️ Views: 314
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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}
\mylink \hfill \mylink\\
\exe{1}(4.5pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
&Soit le polynome \(P(x)=x^3-2x^2-5x+6\)\\
\phantom{\(\times 33\)}0.5&\mylabel[green]{1} Montrer que \(P(x)\) est divisible par \(x-1\)\\
3&\mylabel[green]{2} Ecrir \(P(x)\) sous forme de produit de binomes\\
1&\mylabel[green]{3} Résoudre \(x\in \mathbb{R}\quad P(x) < 0\)\\
\end{tabular}
\\
\exe{2}(11.5pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
\(1.5\times3\)&\mylabel[green]{1.a} Placer les points \(A\left(\frac{-23\pi}{3}\right)\),
\(B\left(\frac{25\pi}{2}\right)\) et \(C\left(\frac{10\pi}{3}\right)\) sur
le cercle trigonométrique\\
2& \mylabel[green]{1.b} Determiner les mesures
\phantom{aa} principales \phantom{aa} des \phantom{aa} angles
\(\left(\widehat{\overrightarrow{OI},\overrightarrow{OA}}\right)\) et
\(\left(\widehat{\overrightarrow{OB},\overrightarrow{OC}}\right)\)\\
1& \mylabel[green]{2.a} Sachant \phantom{aa} que \phantom{aa} \(\cos \frac{\pi}{8} = \frac{\sqrt{2+\sqrt2}}{2}\).
Determiner \(\sin \frac{\pi}{8}\)\\
2& \mylabel[green]{2.b} En déduir : \(\sin \frac{7\pi}{8}\) et \(\sin \frac{3\pi}{8}\)\\
2& \mylabel[green]{3} Simplifier puis calculer:
\(A= \cos \frac{\pi}{5} +\cos \frac{2\pi}{5} +\cos \frac{3\pi}{5} +\cos \frac{4\pi}{5}\)\\
\end{tabular}
\\
\exe{3}(4pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
\(1.5 \times 2\)&\mylabel[green]{1} Résoudre \(x\in [0,2\pi]\quad 2\sin x-1 = 0\) et \(x\in [0,2\pi]\quad 2\cos x+\sqrt3 = 0\)\\
1&\mylabel[green]{2} Résoudre \(x\in [0,2\pi]\quad (2\sin x-1)(2\cos x+\sqrt3) > 0\)\\
\end{tabular}
\mylink \hfill \mylink\\
\\[2cm]
%===========================================================================
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\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}(4.5pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
&Soit le polynome \(P(x)=-2x^3-x^2+8x+4\).\\
\phantom{\(\times 33\)}0.5&
\mylabel[green]{1} Vérifiez que \(-2\) est une racine du polynome \(P(x)\)\\
3&\mylabel[green]{2} Ecrir \(P(x)\) sous forme de produit de binomes\\
1&\mylabel[green]{3} Résoudre \(x\in \mathbb{R}\quad P(x) \ge 0\)\\
\end{tabular}
\\
\exe{2}(11.5pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
\(1\)&\mylabel[green]{1} Placer le point \(A\left(\frac{-121\pi}{3}\right)\),
sur le cercle trigonométrique\\
2& \mylabel[green]{2} Sachant \phantom{aa} que \phantom{aa} \(\cos \frac{\pi}{8} = \frac{\sqrt{2+\sqrt2}}{2}\).
Determiner \(\cos \frac{-\pi}{8}\) et \(\sin \frac{-\pi}{8}\)\\
&\mylabel[green]{3} Soient
\(A(x)=\sin\left(\frac{17\pi}{2}+x\right)\cdot\tan(\pi+x)+\cos\left(\frac{\pi}{2}-x)\cdot\tan\left(\frac{\pi}{2}+x\right)\)\\
& \phantom{\mylabel[green]{3} Soient} \(B(x)=\tan\left(\frac{13\pi}{2}-x\right)\cdot\sin(\pi-x)+\sin\left(\frac{\pi}{2}+x\right)\cdot\tan(13\pi+x)\)\\
4 & \mylabel[lightblue]{3.1} Montrer que \(A(x)=\sin x-\cos x \quad\) et \(\quad B(x)=\sin x +\cos x\)\\
1 & \mylabel[lightblue]{3.2} Montrer que \(A(x)\times B(x)=1-2\cos^2 x\)\\
\phantom{\(\times 33\)}1.5& \mylabel[lightblue]{3.3} Calculer \(\cos x\)
sachant que \(A(x)=\frac{\sqrt3+1}{2}\) et \(B(x)=\frac{\sqrt3-1}{2}\)
et \(x\in ]\frac{\pi}{2},\pi]\)\\
2& \mylabel[lightblue]{3.4} En déduir \(\sin x\) et \(\tan x\)
\end{tabular}
\\
\exe{2}(4pts)\\
\noindent
\begin{tabular}{@{}>{\hfill}r|p{0.96\textwidth}@{}}
\(1.5 \times 2\)&\mylabel[green]{1} Résoudre \(x\in [0,2\pi]\quad 2\sin x+1 = 0\) et \(x\in [0,2\pi]\quad 2\cos x+\sqrt2 = 0\)\\
1&\mylabel[green]{2} Résoudre \(x\in [0,2\pi]\quad (2\sin x+1)(2\cos x+\sqrt2) \le 0\)\\
\end{tabular}
\mylink \hfill \mylink\\
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