Complexes + Trigonométrie

Question 1

définition imaginaire pur

Answer 1

Re(z)=0

Question 2

formule module de z + propriétés

Answer 2

lzl=√(Re(z)^2 + Im(z)^2) = lzbarrel
lzl = √(zzbarre)
lz1z2l = lz1l*lz2l

Question 3

forme exponentielle / forme algébrique d’un complexe

Answer 3

rexp(iθ) / r(cos(θ) + isin(θ))

Question 4

formule de Moivre

Answer 4

(exp(iθ))^n = exp( in*θ)

Question 5

formules d’Euler

Answer 5

cos(θ)= (exp(iθ)+exp( -iθ)) /2

sin(θ)=(exp(iθ) - exp( -iθ) ) / 2i

Question 6

formules trigonométriques:
cos(a+b)
sin(a+b)
cos(a-b)
sin(a-b)
sin(pi-θ)
sin(pi+θ)
cos(pi-θ)
cos(pi+θ)
sin(pi/2 - θ)
sin(pi/2 + θ)
cos(pi/2 - θ)
cos(pi/2 +θ)
cos^2(θ) + sin^2(θ) 
sin(2θ)
cos(2θ)

Answer 6

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)
sin(a+b)=cos(a)sin(b)+cos(b)sin(a)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
sin(a-b)=cos(a)sin(b)-cos(b)sin(a)
sin(pi-θ)=sin(θ)
sin(pi+θ)= -sin(θ)
cos(pi-θ)= -cos(θ)
cos(pi+θ)= -cos(θ)
sin(pi/2 - θ)=cos(θ)
sin(pi/2 + θ)=cos(θ)
cos(pi/2 - θ)=sin(θ)
cos(pi/2 +θ)=-sin(θ)
cos^2(θ) + sin^2(θ) = 1
sin(2θ)= 2 cos(θ)sin(θ)
cos(2θ)= cos^2(θ) - sin^2(θ) = 2 cos^2(θ) - 1 = 1-2sin^2(θ)

Question 7

Inégalité triangulaire

Answer 7

llz1l-lz2ll ≤ lz1+z2l ≤ lz1l + lz2l