Soit les nombres complexes z= 1-i√3 et z2 =e^3i PIE/ 4Determiner la forme algebrique de z2 calculez .... Pergunta de ideia deJis

April 2019 1 146 Report

Soit les nombres complexes z= 1-i√3 et z2 =e^3i PIE/ 4
Determiner la forme algebrique de z2
calculez la forme algebrique de z1 * z2
Ecrire z1 sous forme exponentielle
Calculer z1 * z2 sous forme exponentielle
deduire des questions la valeur exacte de cos 5 PIE/ 12 et de sin 5 PIE/12

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Commentaires Bonsoir,

1) $z_2=e^{3i\frac{\pi}{4}}=cos(\dfrac{3\pi}{4})+isin(\dfrac{3\pi}{4})=-\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2}$

2) $z_1\times z_2=(1-i\sqrt{3})(-\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2})=-\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{6}i}{2}+\dfrac{\sqrt{6}}{2}\\\\=\dfrac{\sqrt{6}-\sqrt{2}}{2}+\dfrac{\sqrt{6}+\sqrt{2}}{2}i$

3) $1-i\sqrt{3}=r\times cis(\theta)\ \ avec\ r = \sqrt{1^2+(\sqrt{3})^2}=\sqrt{1+3}=2\\\\\theta=tan^{-1}(-\dfrac{\sqrt{3}}{1})=-\dfrac{\pi}{3}\\\\\\1-i\sqrt{3}=2e^{-i\dfrac{\pi}{3}}$

4) $z_1\times z_2=2e^{-i\dfrac{\pi}{3}}\times e^{i\dfrac{3\pi}{4}}=2e^{i(-\dfrac{\pi}{3}+\dfrac{3\pi}{4})}\\\\=2e^{i(-\dfrac{4\pi}{12}+\dfrac{9\pi}{12})}=2e^{i\dfrac{5\pi}{12}}$

5) $z_1\times z_2= 2e^{i\dfrac{5\pi}{12}}=2(cos(\dfrac{5\pi}{12})+isin(\dfrac{5\pi}{12}))\\\\=2cos(\dfrac{5\pi}{12})+2isin(\dfrac{5\pi}{12})\\\\\\z_1\times z_2=\dfrac{\sqrt{6}-\sqrt{2}}{2}+\dfrac{\sqrt{6}+\sqrt{2}}{2}i$

En identifiant les parties réelles et imaginaires, nous en déduisons :

$2cos(\dfrac{5\pi}{12})=\dfrac{\sqrt{6}-\sqrt{2}}{2}\\\\cos(\dfrac{5\pi}{12})=\dfrac{\sqrt{6}-\sqrt{2}}{4}\\\\\\2sin(\dfrac{5\pi}{12})=\dfrac{\sqrt{6}+\sqrt{2}}{2}\\\\sin(\dfrac{5\pi}{12})=\dfrac{\sqrt{6}+\sqrt{2}}{4}$

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