TS – Complexes3 – Ex 3

Exercice 3

Mettre chaque nombre complexe sous forme trigonométrique.

  1. $z = (-1 + \ic)^5$
    $\quad$
  2. $z = \left(\sqrt{3} – \ic\right)^4$
    $\quad$
  3. $z = \dfrac{\left(\sqrt{2} – 1\right)\ic}{1 – \ic}$

Correction

  1. $|- 1 + \ic| = \sqrt{2}$
    Donc $-1 + \ic = \sqrt{2} \left(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}\right) = \sqrt{2}\left(\cos \dfrac{3\pi}{4} + \ic \sin \dfrac{3\pi}{4}\right)$.
    Donc arg$(-1 + \ic) = \dfrac{3\pi}{4} \quad (2\pi)$.
    Par conséquent arg$\left((-1 + \ic)^5\right) = 5 \times \dfrac{3\pi}{4} \quad (2\pi) = -\dfrac{\pi}{4} \quad (2\pi)$
    $\quad$
    Ainsi
    $\begin{align} (-1 + \ic)^5 &= \sqrt{2}^5\left(\cos \dfrac{-\pi}{4}+\ic \sin \dfrac{\pi}{4}\right) \\\\
    &= 4\sqrt{2}\left(\cos \dfrac{-\pi}{4}+\ic \sin \dfrac{\pi}{4}\right)
    \end{align}$
    $\quad$
  2. $\left|\sqrt{3} – \ic \right| = 2$.
    $\sqrt{3} – \ic = 2 \left(\dfrac{\sqrt{3}}{2} – \dfrac{\ic}{2}\right) = 2\left(\cos \dfrac{-\pi}{6} + \ic \sin \dfrac{-\pi}{6}\right)$
    Donc arg$\left(\sqrt{3} – \ic\right) = -\dfrac{\pi}{6} \quad (2\pi)$.
    $\quad$
    Par conséquent arg$\left(\left(\sqrt{3} – \ic\right)^4\right) = 4 \times \dfrac{-\pi}{6} = -\dfrac{2\pi}{3} \quad (2\pi)$.
    $\quad$
    Ainsi
    $\begin{align} \left(\sqrt{3} – \ic\right)^4 &= 2^4\left(\cos \dfrac{-2\pi}{3} + \ic \sin \dfrac{-2\pi}{3} \right) \\\\
    & = 16\left(\cos \dfrac{-2\pi}{3} + \ic \sin \dfrac{-2\pi}{3} \right)
    \end{align}$
    $\quad$
  3. $\left|\left(\sqrt{2} – 1\right)\ic\right| = \sqrt{2} – 1$ $\quad$ arg$\left(\left(\sqrt{2} – 1\right)\ic\right) = \dfrac{\pi}{2}$.
    $\quad$
    $|1 – \ic| = \sqrt{2}$
    $1 – \ic| = \sqrt{2}\left(\dfrac{1}{\sqrt{2}} – \dfrac{\ic}{\sqrt{2}}\right) = \sqrt{2}\left(\cos \dfrac{-\pi}{4}+\ic \sin \dfrac{-\pi}{4}\right)$
    Donc arg $(1 – \ic) = -\dfrac{-\pi}{4}$
    $\quad$
    Ainsi $|z| = \dfrac{\sqrt{2} – 1}{\sqrt{2}}$
    Et arg$(z) = \dfrac{\pi}{2} – \dfrac{-\pi}{4} = \dfrac{3\pi}{4}$
    $\quad$
    Donc $z = \dfrac{\sqrt{2} – 1}{\sqrt{2}}\left(\cos \dfrac{3\pi}{4} + \ic \sin \dfrac{3\pi}{4} \right)$