Cercle trigonométrique – exercices de maths 1ère

Exercice 1Conversions

1. \(60^\circ\), \(135^\circ\), \(150^\circ\), \(240^\circ\), \(330^\circ\) en radians.
2. \(\dfrac{\pi}{4}\), \(\dfrac{2\pi}{3}\), \(\dfrac{7\pi}{4}\), \(\dfrac{11\pi}{6}\) en degrés.

Exercice 2Placer

\(\dfrac{\pi}{6}\), \(-\dfrac{\pi}{3}\), \(\dfrac{3\pi}{4}\), \(\dfrac{5\pi}{3}\), \(-\pi\), \(-\dfrac{5\pi}{6}\), \(\dfrac{9\pi}{4}\).

Exercice 3Mesure principale

1. \(\dfrac{11\pi}{4}\)
2. \(\dfrac{19\pi}{6}\)
3. \(-\dfrac{8\pi}{3}\)
4. \(\dfrac{25\pi}{2}\)
5. \(75\pi\)

Exercice 4Valeurs

1. \(\cos\dfrac{\pi}{6}\), \(\sin\dfrac{\pi}{6}\)
2. \(\cos\dfrac{2\pi}{3}\), \(\sin\dfrac{2\pi}{3}\)
3. \(\cos\dfrac{7\pi}{6}\), \(\sin\left(-\dfrac{\pi}{4}\right)\)
4. \(\cos\dfrac{5\pi}{3}\), \(\sin\dfrac{5\pi}{3}\)

Exercice 5Associés

1. \(\cos(-x)\), \(\sin(-x)\)
2. \(\cos(\pi - x)\), \(\sin(\pi - x)\)
3. \(\cos(\pi + x)\), \(\sin(\pi + x)\)
4. \(\cos\left(\dfrac{\pi}{2} + x\right)\), \(\sin\left(\dfrac{\pi}{2} - x\right)\)

Exercice 6Relation

1. \(\sin x = \dfrac{4}{5}\), \(x \in \left[\dfrac{\pi}{2}\,;\,\pi\right]\) : \(\cos x\) ?
2. \(\cos x = -\dfrac{2}{5}\), \(x \in [\pi\,;\,2\pi]\) : \(\sin x\) ?
3. \((\cos x + \sin x)(\cos x - \sin x)\) ?

Exercice 7Équations

1. \(\cos x = -\dfrac{1}{2}\)
2. \(\sin x = \dfrac{\sqrt{2}}{2}\)
3. \(\sin x = 1\)
4. \(\cos x = \dfrac{\sqrt{3}}{2}\)
5. \(2\sin x + \sqrt{3} = 0\)

Dans \(]-\pi\,;\,\pi]\) puis \(\mathbb{R}\).

Exercice 8Inéquations

1. \(\cos x \leq \dfrac{1}{2}\)
2. \(\sin x > -\dfrac{\sqrt{2}}{2}\)
3. \(\sin x < 0\)

Dans \([0\,;\,2\pi[\).

Exercice 9Fonctions

1. Parité, période ?
2. Courbes sur \([-\pi\,;\,2\pi]\).
3. Variations de \(\cos\) sur \([0\,;\,2\pi]\) ?

Exercice 10Problème

\(f(x) = 2\sin x \cos x\).

1. \(f(0)\), \(f\!\left(\dfrac{\pi}{4}\right)\), \(f\!\left(\dfrac{\pi}{2}\right)\) ?
2. \(f(x) = \sin(2x)\) (admis) : période ?
3. \(f(x) = 0\) dans \([0\,;\,2\pi[\) ?