Complexes

Question 1

e(0) =exp(2iπ) =

Answer 1

1

Question 2

exp(iπ) =

Answer 2

−1

Question 3

exp(iπ/2) =

Answer 3

i

Question 4

exp(iπ/4) =

Answer 4

(1+i)/√2

Question 5

exp(iπ/3) =

Answer 5

(1+i√3)/2

Question 6

exp(iπ/6) =

Answer 6

(i+√3)/2

Question 7

exp(2iπ/3) =

Answer 7

(-1+i√3)/2

Question 8

U :

Answer 8

ensemble des complx de module 1, est aussi l’ensemble des complexes de la forme eiθ, θ ∈ R

Question 9

z ∈ U ⇔

Answer 9

z ̄ = 1 /z

Question 10

(cosθ + isinθ)^n =

Answer 10

cosnθ + isinnθ

Question 11

Euler : cos θ =

Answer 11

(eiθ + e−iθ)/2

Question 12

Euler : sin θ =

Answer 12

(eiθ - e−iθ)/2i

Question 13

Formule algébrique : cosθ =

Answer 13

Rez = x

|z| √( x^2 + y^2)

Question 14

Formule algébrique : sinθ =

Answer 14

Imz = y

|z| √( x^2 + y^2)

Question 15

Formule algébrique : tanθ =

Answer 15

Imz = y

Rez x

Question 16

j racine cubique de l’unité (j^3 = 1)

Answer 16

exp(2iπ/3) = (-1+i√3)/2

Question 17

1 + j + j^2 =

Answer 17

0

Question 18

Une racine n-ième de l’unité est un nombre complexe z vérifie :

Answer 18

z^n = 1 donc z = exp((2ikπ)/n)
k parcourant [[0, n − 1]]
et ω = e(2iπ/n)

Question 19

La somme des racines n-iems de l’unité est …

1 + ω + ω^2 + · · · + ω^n−1 =

Answer 19

nulle

0

Question 20

1 + exp(iθ) =

Answer 20

2cos(θ/2)exp(iθ/2)

Question 21

1 - exp(iθ) =

Answer 21

-2isin(θ/2)exp(iθ/2)