Bonjour;
1) soit z = √3/2 - 3i/2
|z| = √(3/4 + 9/4) = √3
⇒ z = √3[1/2 - i√3/2]
⇒ arg (z) = -π/3
⇒ z = √3e^(-iπ/3)
⇒ z¹² = (√3)¹² x [e^(-iπ/3)¹²
⇔ z¹² = 729 x e^(-4iπ)
et e^(-4π) = cos(-4π) + isin(-4π) = 1
donc z¹² = 729
2)
a) e^(iπ/3) x e^(-iπ/4) = e^i(π/3 - π/4) = e^(iπ/12)
b) e^(iπ/12) = cos(π/12) + isin(π/12)
Soit z = e^(iπ/3) = 1/2 + i√3/2
et z' = e^(-iπ/4) = √2/2 - i√2/2
⇒ zz' = (1/2 + i√3/2)(√2/2 - i√2/2) = √2/4 - i√2/4 + i√3√2/4 + √3√2/4
= √2/4(1 + √3) - i√2/4(1 + √3)
= (1 + √3)√2/4 * (1 - i)
⇒ |zz'| = (1 + √3)√2/4 * √2 = (1 + √3)/2
⇒ zz' = (1 + √3)/2 * [√2/2 - i√2/2] = |zz'| * [(cos(-π/4) + isin(-π/4)] = |zz'|e^(-iπ/4)
Or zz' = cos(π/12) + isin(π12)
⇒ cos(π12) = |zz'| * cos(-π/4) = (1 + √3)/2 * √2/2 = √2(1 + √3)/4 = (1 + √3)/2√2
