Bonjour,
1) (1 + i)⁴ⁿ
= [√2(√2/2 + i√2/2)]⁴ⁿ
= [√2e(iπ/4)]⁴ⁿ
= (√2)⁴ⁿ x (e(iπ/4))⁴ⁿ
= 4ⁿ x (e(iπ))ⁿ
= 4ⁿ x (-1)ⁿ
= (-4)ⁿ
2) Les solutions de (E) sont :
a =4
b = 2 - 2i
c = 2 + 2i
Faire une figure et calculer les distances1
Aire = 8
3)
1 + e(2αi)
= 1 + (e(αi)²
= 1 + [cos(α) + isin(α)]²
= 1 + cos²(α) + 2isin(α)cos(α) - sin²(α)
= 1 + cos²(α) + 2isin(α)cos(α) - 1 + cos²(α)
= 2cos(α)[cos(α) + isin(α)]
= 2cos(α) x e(iα)
4) arg(zA) = π/4
arg[(zA)ⁿ] = n x arg(zA) = n x π/4
O, A et Mn si et seulement si :
n x π/4 = π/4 + kπ
⇔ n = 1 + 4k
⇔ n - 1 = 4k
5) j = cos(2π/3) + isin(2π/3)
= -1/2 + i√3/2
j² = 1/4 - i√3/2 - 3/4 = -1/2 - i√3/2
⇒ 1 + j + j² = 1 - 1/2 + i√3/2 - 1/2 - i√3/2 = 0
