✅ Corrigé détaillé — Exercice 2
Partie A
| Expression | Formule d’angle associé | Valeur exacte |
|---|---|---|
| a) cos(2π/3) | cos(π − π/3) = −cos(π/3) | −1/2 |
| b) sin(5π/6) | sin(π − π/6) = sin(π/6) | 1/2 |
| c) cos(−π/3) | cos(−x) = cos(x) → cos(π/3) | 1/2 |
| d) sin(7π/6) | sin(π + π/6) = −sin(π/6) | −1/2 |
| e) cos(5π/4) | cos(π + π/4) = −cos(π/4) | −√2/2 |
| f) sin(11π/6) | sin(2π − π/6) = sin(−π/6) = −sin(π/6) | −1/2 |
Partie B
2a) cos(π−x) = −cos(x) = −1/3.
2b) sin(π+x) = −sin(x) = −(−2√2/3) = 2√2/3.
2c) cos(−x) = cos(x) = 1/3.
2d) sin(π/2 − x) = cos(x) = 1/3.
Partie C
3a) sin(100π + π/3) = sin(π/3 + 100π) = sin(π/3 + 50 × 2π) = sin(π/3) = √3/2.
(100π = 50 × 2π, donc on effectue 50 tours entiers).
3b) cos(−7π + π/4) = cos(−7π + π/4 + 4×2π) = cos(π + π/4) = −cos(π/4) = −√2/2.
(−7π + 8π = π, puis cos(π + π/4) = −cos(π/4).)
3c) sin(5π/3 + 8π) = sin(5π/3 + 4 × 2π) = sin(5π/3) = sin(2π − π/3) = sin(−π/3) = −sin(π/3) = −√3/2.