Équations (2) - Corrigé

Énoncé

Résoudre dans \(\mathbb{C}\) les équations suivantes.

1. \(z^4=(2-i)^4\)

2. \(z^5= \sqrt{3}+3i\)

3. \(z^6= \text e^{\frac{i \pi}{7}}\)

Solution

1. Pour tout \(z \in \mathbb{C}\) ,
\(\begin{align*}z^4= (2-i)^4& \Longleftrightarrow \frac{z^4}{(2-i)^4} = 1\\& \Longleftrightarrow \left( \frac{z}{2-i} \right)^4 = 1\\& \Longleftrightarrow \frac{z}{2-i} \in \mathbb{U}_4\\& \Longleftrightarrow \frac{z}{2-i} \in \left\lbrace 1 \ ; i \ ; -1 \ ; -i \right\rbrace\\& \Longleftrightarrow z \in \left\lbrace 2-i \ ; i \times (2-i) \ ; - (2-i) \ ; -i\times (2-i) \right\rbrace\end{align*}\)
donc \(S= \left\lbrace 2-i \ ; 1+2i \ ; -2+i \ ; -1-2i \right\rbrace\) .

2. \(\sqrt{3}+3i = 2 \sqrt{3} \text e^{\frac{i\pi}{6}}\)
Pour tout \(z \in \mathbb{C}\) ,
\(\begin{align*}z^5= \sqrt{3}+3i& \Longleftrightarrow z^5= 2 \sqrt{3} \text e^{\frac{i\pi}{6}}\\ & \Longleftrightarrow z^5= \left( \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}} \right)^5\\ & \Longleftrightarrow \frac{z^5}{ \left( \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}} \right)^5 } = 1\\& \Longleftrightarrow \left( \frac{z}{\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}} \right)^5 = 1\\& \Longleftrightarrow \frac{z}{\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}} \in \mathbb{U}_5\\& \Longleftrightarrow \frac{z}{\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}} \in \left\lbrace 1 \ ; \text e^{\frac{2i\pi}{5}} \ ;\text e^{\frac{4i\pi}{5}} \ ;\text e^{\frac{6i\pi}{5}} \ ; \text e^{\frac{8i\pi}{5}} \right\rbrace\\& \Longleftrightarrow z \in \left\lbrace \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}} \ ; \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}} \text e^{\frac{2i\pi}{5}} \ ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}\text e^{\frac{4i\pi}{5}} \ ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}e^{\frac{6i\pi}{5}} \ ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}}\text e^{\frac{8i\pi}{5}} \right\rbrace\\& \Longleftrightarrow z \in \left\lbrace\sqrt[5]{2} \sqrt[10]{3}\text e^{\frac{i\pi}{30}} \ ; \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{13i\pi}{30}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{5i\pi}{6}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{37i\pi}{30}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{49i\pi}{30}} \right\rbrace\end{align*}\)
donc \(S= \left\lbrace\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{i\pi}{30}} \ ; \sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{13i\pi}{30}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{5i\pi}{6}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{37i\pi}{30}} ;\sqrt[5]{2} \sqrt[10]{3} \text e^{\frac{49i\pi}{30}} \right\rbrace\) .

3. Pour tout \(z \in \mathbb{C}\) ,
\(\begin{align*}z^6= \text e^{\frac{i\pi}{7}}& \Longleftrightarrow z^6 = \left( \text e^{\frac{i\pi}{42}} \right)^6\\ & \Longleftrightarrow \frac{z^6}{ \left( \text e^{\frac{i\pi}{42}} \right)^6 } = 1\\& \Longleftrightarrow \left( \frac{z}{ \text e^{\frac{i\pi}{43}}} \right)^6 = 1\\& \Longleftrightarrow \frac{z}{\text e^{\frac{i\pi}{42}}} \in \mathbb{U}_6\\& \Longleftrightarrow \frac{z}{ \text e^{\frac{i\pi}{42}}} \in \left\lbrace 1 \ ; \text e^{\frac{i\pi}{3}} \ ; \text e^{\frac{2i\pi}{3}} \ ; -1 \ ; \text e^{\frac{4i\pi}{3}} \ ; \text e^{\frac{5i\pi}{3}} \right\rbrace\\& \Longleftrightarrow z \in\left\lbrace \text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{i\pi}{42}}\text e^{\frac{i\pi}{3}} \ ; \text e^{\frac{i\pi}{42}}\text e^{\frac{2i\pi}{3}} \ ;-\text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{i\pi}{42}}\text e^{\frac{4i\pi}{3}} \ ; \text e^{\frac{i\pi}{42}}\text e^{\frac{5i\pi}{3}} \right\rbrace\\& \Longleftrightarrow z \in\left\lbrace \text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{5i\pi}{14}};\text e^{\frac{29i\pi}{42}} \ ;-\text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{19i\pi}{14}};\text e^{\frac{71i\pi}{42}} \right\rbrace\end{align*}\)

\(S = \left\lbrace \text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{5i\pi}{14}};\text e^{\frac{29i\pi}{42}} \ ;-\text e^{\frac{i\pi}{42}} \ ;\text e^{\frac{19i\pi}{14}};\text e^{\frac{71i\pi}{42}} \right\rbrace\) .