✅ Corrigé détaillé — Exercice 3
Partie A
| Expression | Développement | Valeur |
|---|---|---|
| a) cos(3π/4) + sin(3π/4) | cos(π−π/4) + sin(π−π/4) = −cos(π/4) + sin(π/4) = −√2/2 + √2/2 | 0 |
| b) cos²(π/6) + sin²(π/3) | (√3/2)² + (√3/2)² = 3/4 + 3/4 | 3/2 |
| c) 2sin(5π/6) − cos(4π/3) | 2×sin(π−π/6) − cos(π+π/3) = 2×(1/2) − (−1/2) = 1 + 1/2 | 3/2 |
Partie B
2a) cos²(x) + sin²(x) = 1 → sin²(x) = 1 − 4/9 = 5/9.
sin(x) < 0 → sin(x) = −√5/3.
2b) • sin(π + x) = −sin(x) = √5/3.
• cos(π − x) = −cos(x) = 2/3.
• sin(−x) = −sin(x) = √5/3.
• cos(π/2 − x) = sin(x) = −√5/3.
Partie C
3) sin(x) = √3/2. Les angles de [0 ; π] dont le sinus vaut √3/2 : x = π/3 et x = π − π/3 = 2π/3. ✓
Solutions dans [0 ; 2π] : x = π/3 et x = 2π/3.
4) cos(x) = −1/2. Les angles de [0 ; 2π] dont le cosinus vaut −1/2 :
cos(π/3) = 1/2 → cos(π − π/3) = −1/2 → x = 2π/3.
cos(2π − π/3) = cos(−π/3) = 1/2 → cos(2π − 2π/3) = cos(4π/3) = −1/2 → x = 4π/3.
Solutions : x = 2π/3 et x = 4π/3.
5) sin(x) ≥ 1/2 sur [0 ; 2π]. sin(x) = 1/2 pour x = π/6 et x = 5π/6. Le sinus est ≥ 1/2 entre ces deux valeurs (dans le demi-cercle supérieur).
S = [π/6 ; 5π/6].