z1=1-i=√2(√2/2-√2/2.i)=√2.exp(-π/4.i)
z2=-8-8√3.i=16(-1/2-√3/2.i)=16.exp(-2π/3.i)
Z=z1/z2=√2/16.exp((-π/4+2π/3).i)=√2/16.exp(5π/12.i)
par ailleurs
Z=z1/z2=(1-i)/(-8-8√3i)
=((1-i)(-8+8√3i))/((-8)²+(-8√3)²)
=(-8+8√3i+8i+8√3)/(256)
=(-1+√3)/32+i.(1+√3)/32
donc √2/16.cos(5π/12)=(-1+√3)/32
donc cos(5π/12)=(-1+√3)/(2√2)=(√6-√2)/4
de même : √2/16.sin(5π/12)=(1+√3)/32
donc sin(5π/12)=(1+√3)/(2√2)=(√6+√2)/4
enfin l'équation est : (E) :(√6-√2).cos(x)-(√6+√2).sin(x)=-2√3
donc (√6-√2)/4.cos(x)-(√6+√2)/4.sin(x)=-√3/2
donc cos(5π/12).cos(x)-sin(5π/12).sin(x)=cos(5π/6)
donc cos(x+5π/12)=cos(5π/6)
donc : x+5π/12=5π/6+2kπ ou x+5π/12=-5π/6+2kπ
donc x=5π/12+2kπ ou x=-5π/4+2kπ
