Complexes de module 1 - Corrigé

Énoncé

Montrer que les complexes suivants appartiennent à  \(\mathbb{U}\) .

• \(z_1=\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i\)
  
• \(z_2=\dfrac{2}{5}+\dfrac{\sqrt{21}}{5}i\)
  
• \(z_3=i\)

Solution

\(\begin{align*}\lvert z_1 \lvert& = \lvert \frac{1}{2}-\frac{\sqrt{3}}{2}i \lvert= \sqrt{\left(\frac{1}{2}\right)^2+\left(-\frac{\sqrt{3}}{2}\right)^2}= \sqrt{\frac{1}{4}+\frac{3}{4}}= \sqrt{\frac{4}{4}}= \sqrt{1}= 1\end{align*}\)
Donc  \(z_1 \in \mathbb{U}\) .

\(\begin{align*}\lvert z_2 \lvert& = \lvert \frac{2}{5}+\frac{\sqrt{21}}{5}i \lvert= \sqrt{\left(\frac{2}{5}\right)^2+\left(\frac{\sqrt{21}}{5}\right)^2}= \sqrt{\frac{4}{25}+\frac{21}{25}}= \sqrt{\frac{25}{25}}= \sqrt{1}= 1\end{align*}\)
Donc  \(z_2 \in \mathbb{U}\) .

\(\begin{align*}\lvert z_3 \lvert& = \lvert i \lvert= \sqrt{0^2+1^2}= \sqrt{1}= 1\end{align*}\)
Donc  \(z_3 \in \mathbb{U}\) .