Exercice 1
- Convertir en radians : \(150^{\circ}\), \(210^{\circ}\), \(-45^{\circ}\).
- Déterminer l'abscisse curviligne principale de \(\frac{17\pi}{3}\) et de \(-\frac{29\pi}{4}\).
Correction
- \(\frac{5\pi}{6}\), \(\frac{7\pi}{6}\), \(-\frac{\pi}{4}\).
- \(\frac{17\pi}{3} = 6\pi - \frac{\pi}{3}\) : principale \(-\frac{\pi}{3}\). \(-\frac{29\pi}{4} = -8\pi + \frac{3\pi}{4}\) : principale \(\frac{3\pi}{4}\).
Exercice 2
Calculer : \(\cos\frac{5\pi}{6}\), \(\sin\frac{7\pi}{6}\), \(\cos\frac{2\pi}{3}\), \(\tan\frac{3\pi}{4}\).
Correction
\(\cos\frac{5\pi}{6} = \cos(\pi - \frac{\pi}{6}) = -\frac{\sqrt3}{2}\). \(\sin\frac{7\pi}{6} = \sin(\pi + \frac{\pi}{6}) = -\frac12\). \(\cos\frac{2\pi}{3} = -\frac12\). \(\tan\frac{3\pi}{4} = -1\).
Exercice 3
Sachant que \(\sin x = \frac35\) et \(x \in [\frac{\pi}{2} ; \pi]\), calculer \(\cos x\) et \(\tan x\).
Correction
\(\cos^2 x = 1 - \frac{9}{25} = \frac{16}{25}\) et \(\cos x \le 0\) sur cet intervalle : \(\cos x = -\frac45\). \(\tan x = \frac{3/5}{-4/5} = -\frac34\).
Exercice 4
Simplifier \(A = \cos(\pi - x) + \sin(\frac{\pi}{2} - x) + \cos(\pi + x) + \cos(-x)\).
Correction
\(A = -\cos x + \cos x - \cos x + \cos x = 0\).
Exercice 5
Résoudre dans \(\mathbb{R}\) puis dans \(]-\pi ; \pi]\) :
- \(\sin x = \frac{\sqrt2}{2}\)
- \(2\cos x + \sqrt3 = 0\)
Correction
- \(x = \frac{\pi}{4} + 2k\pi\) ou \(x = \frac{3\pi}{4} + 2k\pi\). Dans \(]-\pi;\pi]\) : \(\{\frac{\pi}{4} ; \frac{3\pi}{4}\}\).
- \(\cos x = -\frac{\sqrt3}{2} = \cos\frac{5\pi}{6}\) : \(x = \pm\frac{5\pi}{6} + 2k\pi\). Dans \(]-\pi;\pi]\) : \(\{-\frac{5\pi}{6} ; \frac{5\pi}{6}\}\).
Exercice 6
Montrer que pour tout \(x\) : \((\cos x + \sin x)^2 + (\cos x - \sin x)^2 = 2\).
Correction
En développant : \((\cos^2x + 2\cos x\sin x + \sin^2x) + (\cos^2x - 2\cos x\sin x + \sin^2x) = 2(\cos^2x + \sin^2x) = 2\).