3 cos(32π) +i sin(32π) (1−i) (−3i

5.10 1) |1−i|=p

1²+ (−1)² =√ 2 1−i=√

2

1

√2 −i_√¹

2

=√ 2√

2

2 +i(−^√22)

=√

2 cos(⁷4^π) +i sin(⁷4^π)

−3i= 3 0 +i·(−1)

= 3 cos(³₂^π) +i sin(³₂^π)

(1−i) (−3i) =√

2·3 = 3√ 2 arg (1−i) (−3i)

= ⁷4^π +³2^π = ^13π4 = ⁵4^π + 2π

2) −2i= 2 0 +i·(−1)

= 2 cos(³2^π) +i sin(³2^π)

(−2i)¹⁰

= 2¹⁰= 1024 arg (−2i)¹⁰

= 10·³2^π = 15π=π+ 7·2π 3) |1 +√

3i|= q

1²+ (√

3)² =√

1 + 3 =√ 4 = 2 1 +√

3i= 2 ¹2 +i^√3

2

= 2 cos(^π3) +i sin(^π3)

(1 +√ 3i)²

= 2² = 4 arg (1 +√

3i)²

= 2· ^π3 = ²₃^π 4) | −1 +i|=p

(−1)²+ 1² =√ 2

−1 +i=√

2 −^√12+i_√¹

2

=√

2 −^√22+i

√2 2

=√

2 cos(³4^π) +i sin(³4^π)

|2 + 2i|=|2 (1 +i)|=|2| |1 +i|= 2√

1²+ 1²= 2√ 2 2 + 2i= 2√

2 ₂^√2₂ +i ²

2√ 2

= 2√

2 ^√1₂ +i_√¹

2

= 2√

2 ^√2² +i^√2

2

= 2√

2 cos(^π₄) +i sin(^π₄)

(−1 +i)⁵(2 + 2i)⁴ = (√

2)⁵·(2√

2)⁴ = 4√

2·16·4 = 256√ 2 arg (−1 +i)⁵(2 + 2i)⁴

= 5· ³4^π + 4· ^π4 = ¹⁵₄^π +π= ¹⁹₄^π = ³₄^π + 2·2π 5) |√

3−i|= q

(√

3)²+ (−1)² =√

3 + 1 =√ 4 = 2

√3−i= 2 ^√2³ +i(−¹²)

= 2 cos(⁵6^π) +i sin(⁵6^π)

|√

3 +i|= q

(√

3)²+ 1² =√

3 + 1 =√ 4 = 2

√3 +i= 2 ^√₂³ +i¹

2

= 2 cos(^π₆) +i sin(^π₆)

√3−i

√3 +i

!³⁰

= 2

2 ³⁰

= 1³⁰ = 1

arg





√3−i

√3 +i

!³⁰

= 30· ⁵6^π −^π6

= 30· ²3^π = 20π = 0 + 10·2π

Algèbre : nombres complexes — forme trigonométrique Corrigé 5.10

6) |1−√ 3i|=

q

1²+ (−√

3)² =√

1 + 3 =√ 4 = 2 1−√

3i= 2 ¹₂ +i(−^√2³)

= 2 cos(⁵₃^π) +i sin(⁵₃^π) En 5), on a déjà établi √

3 +i= 2 cos(^π₆) +i sin(^π₆)

1−√ 3i

√3 +i

!¹⁷

= 2

2 ¹⁷

= 1¹⁷ = 1

arg





1−√ 3i

√3 +i

!¹⁷

= 17· ⁵3^π − ^π6

= 17· ³2^π = ⁵¹₂^π = ³₂^π + 12·2π

Algèbre : nombres complexes — forme trigonométrique Corrigé 5.10