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I re C, D – math I – <strong>Les</strong> <strong>nombres</strong> <strong>complexes</strong><br />
π<br />
π i 4<br />
π π 2 2<br />
cis = e = cos + isin = + i⋅<br />
4 4 4 2 2<br />
• Définitions :<br />
<br />
i<br />
z = rcisϕ = re ϕ est appelée : forme trigonométrique de z<br />
z = a + bi est appelée : forme algébrique de z<br />
• Exemples :<br />
π ⎛ 1 3 ⎞<br />
z = 6cis = 6 i 3 3 3 i<br />
3 ⎜<br />
+ = + ⋅<br />
2 2 ⎟<br />
⎝ ⎠<br />
π<br />
i ⎛ 3 1 ⎞ 5 3 5<br />
6<br />
z = 5e = 5 ⎜<br />
+ i = + i<br />
2 2 ⎟<br />
⎝ ⎠ 2 2<br />
c) Propriétés<br />
iϕ<br />
• R<br />
∀ϕ∈ cisϕ = e = 1<br />
en effet<br />
2 2<br />
cisϕ = cos ϕ + sin ϕ = 1 = 1<br />
• cisϕ = cis ϕ' ⇔ ϕ ' = ϕ + 2kπ ( k ∈ Z )<br />
en effet cisϕ = cis ϕ' ⇔ (cos ϕ = cos ϕ' et sin ϕ = sin ϕ ') ,<br />
or cosϕ = cos ϕ' ⇔ ( ϕ = ϕ ' + 2kπ ou ϕ = −ϕ ' + 2kπ<br />
)<br />
et sin ϕ = sin ϕ' ⇔ ( ϕ = ϕ ' + 2kπ ou ϕ = π − ϕ ' + 2kπ ) ,<br />
donc cosϕ = cos ϕ' et sin ϕ = sin ϕ' ⇔ ϕ = ϕ ' + 2kπ.<br />
•<br />
( )<br />
( )<br />
( −ϕ ) =<br />
⎧cis ϕ + 2π = cis ϕ (1)<br />
⎪<br />
∀ϕ∈ R ⎨cis ϕ + π = −cis ϕ (2)<br />
⎪<br />
⎩cis cis ϕ (3)<br />
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