Bonsoir ! J'ai trois exercices sur la forme exponentielle des nombres complexes à faire pour lundi et je galère. Merci d'avance pour votre aide !
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Exo 144 :a) 3 * (cos9o° + i sin9o°) = 3 * i = 3i
b) 5 * (cos(-6o°) + i sin(-6o°)) = 5 * (0,5 - 0,5i√3) = 2,5 - 2,5i√3
c) 4 * e∧i(π/4 - 6π/4) = 4 * e∧i(-5π/4) = 4 * (cos135° + i sin135°)
= 4 * (-0,5√2 + 0,5 i √2)
= -2√2 + 2 i √2
d) 3 * e∧i(π/6 + 4π/6) = 3 * e∧i(5π/6) = 3 * (cos15o° + i sin15o°)
= 3 * (-0,5√3 + 0,5 i )
= -1,5√3 + 1,5 i
exo 145 :
a) 6 = 6 * cos0° = 6 e∧(0i)
b) -8 = 8 * cosπ = 6 e∧(iπ)
c) 1,5 i = 1,5 * (cos9o° + i sin9o°) = 1,5 e∧(iπ/2)
d) (√3 + i)³ = 8 * (0,5√3 + 0,5 i )³ = 8 * (cos3o° + i sin3o°)³
= 8 * [ e∧(iπ/6) ]³
= 8 * e∧(iπ/2)
e) 2 / (1-i) = 2 * e∧0i / [ √2 * e∧(iπ/4) ] = √2 * e∧0i / e∧(iπ/4)
= √2 * e∧(-iπ/4)
exo 146 :
a) 2√3 + 6i = 4√3 * (cos6o° + i sin6o°) = 4√3 e∧(iπ/3)
b) √2 (1+i) e∧(iπ/3) = 2 e∧(iπ/4) * e∧(iπ/3) = 2 e∧(i 7π/12)
remarque : 7π/12 --> 1o5°