Question

calcule la parte real del número complejo :

         e^iα + e^3iα + e^5iα
Z= _________________
        e^2iα + e^6iα + e^10iα

Answer 1

$\text{sea } x=e^{i\alpha}\\ \\ \\ z=\dfrac{x+x^3+x^5}{x^2+x^6+x^{10}}=\dfrac{1+x^2+x^4}{x(1+x^4+x^{8})}\\ \\ \\ z=\dfrac{1}{x(x^4-x^2+1)}=\dfrac{x^2+1}{x(x^6+1)}=\dfrac{x+x^{-1}}{x^3(x^3+x^{-3})}\\ \\ \\ z=\dfrac{\cos\alpha}{\cos3\alpha}x^{-3}=\dfrac{\cos\alpha}{\cos3\alpha}(\cos3\alpha-i\sin3\alpha)\\ \\ \\ \text{Re}(z)=\cos\alpha$