Durée : 1 heure. Barème sur 20.
Exercice 1 (5 pts)
- Convertir \(150°\) en radians et \(\dfrac{7\pi}{4}\) en degrés. (2 pts)
- Donner les valeurs exactes de \(\cos\dfrac{\pi}{3}\), \(\sin\dfrac{\pi}{4}\) et \(\tan\dfrac{\pi}{6}\). (3 pts)
Correction
- \(\dfrac{5\pi}{6}\) ; \(315°\).
- \(\dfrac12\) ; \(\dfrac{\sqrt2}{2}\) ; \(\dfrac{\sqrt3}{3}\).
Exercice 2 (5 pts)
Sachant que \(\sin x = \dfrac{5}{13}\) et \(x\in\left]0\,;\dfrac{\pi}{2}\right[\) :
- Calculer \(\cos x\). (3 pts)
- Calculer \(\tan x\). (2 pts)
Correction
- \(\cos^2 x = 1 - \dfrac{25}{169} = \dfrac{144}{169}\), donc \(\cos x = \dfrac{12}{13}\).
- \(\tan x = \dfrac{5}{12}\).
Exercice 3 (5 pts)
Simplifier \(B = \sin(\pi - x) + \cos\left(\dfrac{\pi}{2} + x\right) + \cos(-x)\).
Correction
\(B = \sin x - \sin x + \cos x = \cos x\).
Exercice 4 (5 pts)
Résoudre dans \([0\,;2\pi[\) :
- \(\sin x = \dfrac{\sqrt3}{2}\) (3 pts)
- \(\cos x = -1\) (2 pts)
Correction
- \(x = \dfrac{\pi}{3}\) ou \(x = \dfrac{2\pi}{3}\).
- \(x = \pi\).