Conforme au programme officiel 2022 - 2023 : infos
z=a+ib=r(cosθ+isinθ)⟹{a=rcosθb=rsinθz=a+ib=r(\cos\theta +i\sin\theta)\Longrightarrow \left\lbrace \begin{array}{l} a=r\cos\theta \ b=r\sin\theta \end{array} \right.z=a+ib=r(cosθ+isinθ)⟹{a=rcosθb=rsinθ
eiθ=cosθ+isinθe^{i\theta}=\cos\theta+i\sin\thetaeiθ=cosθ+isinθ
Calcul du module
r=∣z∣=3+32=3+9=3×4=2×3r =|z| =\sqrt{3+3^2}=\sqrt{3+9} = \sqrt{3 \times 4} = 2 \times \sqrt{3}r=∣z∣=3+32=3+9=3×4=2×3 Calcul de l’argument
{cosθ=ar=32×3=12sinθ=br=−32×3=−3×32×3=−32⟹θ=−π3\left\lbrace \begin{array}{l} \cos\theta = \dfrac a r= \dfrac {\sqrt3}{2 \times \sqrt{3}}= \dfrac 1 2 \ \sin\theta = \dfrac b r = \dfrac {-3}{2 \times \sqrt 3}=\dfrac {-\sqrt 3 \times \sqrt 3}{2 \times \sqrt 3}= -\dfrac{\sqrt 3}{2} \end{array} \right. \Longrightarrow \theta = - \dfrac{\pi}{3}⎩⎪⎪⎨⎪⎪⎧cosθ=ra=2×33=21sinθ=rb=2×3−3=2×3−3×3=−23⟹θ=−3π Écriture sous la forme exponentielle
z=r(cosθ+isinθ)=r×eiθ=2×3×e−iπ3z=r(\cos\theta + i \sin \theta)=r \times e^{i\theta}=2 \times \sqrt 3 \times e^{-\tfrac {i\pi}{3}}z=r(cosθ+isinθ)=r×eiθ=2×3×e−3iπ