Nombres complexes

Sachant que : \(1+ \cos \varphi = 2 \cos^{2} \frac{\varphi}{2}\) et \(\sin \varphi = 2 \cos \frac{\varphi}{2} \sin \frac{\varphi}{2}\) nous obtenons :

\(\underline{z}_{4} =2 \cos^{2} \frac{\varphi}{2} + j2 \cos \frac{\varphi}{2} \sin \frac{\varphi}{2} = 2 \cos \frac{\varphi}{2} \left(\cos \frac{\varphi}{2} + j \sin \frac{\varphi}{2}\right)\)

1er cas : \(\cos \frac{\varphi}{2} \ge 0\)

\(\underline{z}_{4} = 2 \cos \frac{\varphi}{2}\left(\cos \frac{\varphi}{2} + j \sin \frac{\varphi}{2}\right)\)

\(\underline{z}_{4} = 2 \cos \frac{\varphi}{2}e^{\left(j \frac{\varphi}{2}\right)}\) ( 3 points )

2ème cas : \(\cos \frac{\varphi}{2} < 0\)

\(\underline{z}_{4} = -2 \cos \frac{\varphi}{2} \left(\cos \left(\frac{\varphi}{2} + \pi \right) + j \sin \left(\frac{\varphi}{2} + \pi \right)\right)\)

\(\underline{z}_{4} = -2 \cos \frac{\varphi}{2} e^{j\left(\frac{\varphi}{2} + \pi\right)}\) ( 3 points )