Bonjour
1) [tex]Z=\dfrac{1+i\sqrt{3}}{\sqrt{2}-i\sqrt{2}}\\\\Z=\dfrac{(1+i\sqrt{3})(\sqrt{2}+i\sqrt{2})}{(\sqrt{2}-i\sqrt{2})(\sqrt{2}+i\sqrt{2})}\\\\Z=\dfrac{\sqrt{2}+i\sqrt{2}+i\sqrt{6}-\sqrt{6}}{2+2}\\\\Z=\dfrac{(\sqrt{2}-\sqrt{6})+i(\sqrt{2}+\sqrt{6})}{4}\\\\\boxed{Z=\dfrac{\sqrt{2}-\sqrt{6}}{4}+i\dfrac{\sqrt{2}+\sqrt{6}}{4}}[/tex]
2) [tex]z_1=1+i\sqrt{3}\\\\z_1=2(\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2})\\\\z_1=2[\cos(\dfrac{\pi}{3})+i\sin(\dfrac{\pi}{3})]\\\\\boxed{z_1=2e^{i(\dfrac{\pi}{3})}}[/tex]
[tex]z_2=\sqrt{2}-i\sqrt{2}\\\\z_2=2(\dfrac{\sqrt{2}}{2}-i\dfrac{\sqrt{2}}{2})\\\\z_2=2[\cos(\dfrac{\pi}{4})-i\sin(\dfrac{\pi}{4})]\\\\\boxed{z_2=2e^{-i(\dfrac{\pi}{4})}}[/tex]
[tex]Z=\dfrac{z_1}{z_2}\\\\Z=\dfrac{e^{i(\dfrac{\pi}{3})}}{e^{-i(\dfrac{\pi}{4})}}\\\\Z=e^{i(\dfrac{\pi}{3})+i(\dfrac{\pi}{4})}\\\\Z=e^{i(\dfrac{4\pi}{12})+i(\dfrac{3\pi}{12})}\\\\\boxed{Z=e^{i(\dfrac{7\pi}{12})}}[/tex]
3) [tex]Z=e^{i(\dfrac{7\pi}{12})}=\cos(\dfrac{7\pi}{12})+i\sin(\dfrac{7\pi}{12})\\\\Z=\dfrac{\sqrt{2}-\sqrt{6}}{4}+i\dfrac{\sqrt{2}+\sqrt{6}}{4}[/tex]
En comparant les deux écritures de Z, nous en déduisons que :
[tex]\cos(\dfrac{7\pi}{12})=\dfrac{\sqrt{2}-\sqrt{6}}{4}\\\\\sin(\dfrac{7\pi}{12})=\dfrac{\sqrt{2}+\sqrt{6}}{4}[/tex]
4) Constructions en pièces jointe.
5) [tex]Z^{2018}=(e^{i(\dfrac{7\pi}{12})})^{2018}\\\\Z^{2018}=e^{i(\dfrac{7\pi}{12}\times2018)}\\\\Z^{2018}=e^{i(\dfrac{1426\pi}{12})}\\\\Z^{2018}=e^{i(\dfrac{713\pi}{6})}\\\\Z^{2018}=e^{i(\dfrac{708\pi}{6}+\dfrac{7\pi}{6})}[/tex]
[tex]Z^{2018}=e^{i(118\pi+\dfrac{7\pi}{6})}\\\\Z^{2018}=e^{i(59\times2\pi+\dfrac{7\pi}{6})}\\\\Z^{2018}=e^{i(59\times2\pi)}\times e^{i(\dfrac{7\pi}{6})}\\\\Z^{2018}=(e^{i(2\pi)})^{59}\times e^{i(\dfrac{7\pi}{6})}\\\\Z^{2018}=1^{59}\times e^{i(\dfrac{7\pi}{6})}[/tex]
[tex]\\\\Z^{2018}=1\times e^{i(\dfrac{7\pi}{6})}\\\\Z^{2018}=e^{i(\dfrac{7\pi}{6})}\\\\Z=\cos(\dfrac{7\pi}{6})+i\sin(\dfrac{7\pi}{6})\\\\Z=-\dfrac{\sqrt{3}}{2}+i\times(-\dfrac{1}{2})\\\Z=-\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}[/tex]
