QCM trigonometrie 2

1. Quelle est la solution de l’équation \( \cos x = -\frac{1}{2} \) dans l’intervalle \( [-\pi, \pi] \) ?

a) \( x = -\frac{\pi}{3} \)
b) \( x = -\frac{2\pi}{3} \)
c) \( x = -\frac{\pi}{6} \)
d) \( x = -\frac{5\pi}{6} \)

2. Quelle est la solution de l’équation \( \sin x = -\frac{\sqrt{2}}{2} \) dans l’intervalle \( [-\pi, \pi] \) ?

a) \( x = -\frac{\pi}{4} \)
b) \( x = -\frac{3\pi}{4} \)
c) \( x = -\frac{\pi}{6} \)
d) \( x = -\frac{5\pi}{6} \)

3. Quelle est la solution de l’équation \( \tan x = -\frac{\sqrt{3}}{3} \) dans \( \mathbb{R} \) ?

a) \( x = -\frac{\pi}{6} + k\pi \)
b) \( x = -\frac{\pi}{3} + k\pi \)
c) \( x = -\frac{\pi}{4} + k\pi \)
d) \( x = -\frac{\pi}{2} + k\pi \)

4. Quelle est la solution de l’équation \( 2\cos(2x) – 1 = 0 \) dans \( [0, 4\pi] \) ?

a) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)
b) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \)
c) \( x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \)
d) \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2} \)

5. Quelle est la solution de l’équation \( \cos\left(2x + \frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) dans \( \mathbb{R} \) ?

a) \( x = -\frac{\pi}{12} + k\pi \)
b) \( x = -\frac{\pi}{6} + k\pi \)
c) \( x = -\frac{\pi}{4} + k\pi \)
d) \( x = -\frac{\pi}{3} + k\pi \)

6. Quelle est la solution de l’équation \( \sin(2x) = \cos(3x) \) dans \( \mathbb{R} \) ?

a) \( x = \frac{\pi}{10} + \frac{2k\pi}{5} \)
b) \( x = \frac{\pi}{5} + \frac{2k\pi}{5} \)
c) \( x = \frac{\pi}{4} + \frac{2k\pi}{5} \)
d) \( x = \frac{\pi}{2} + \frac{2k\pi}{5} \)

7. Quelle est la solution de l’équation \( \tan\left(2x – \frac{\pi}{4}\right) = -\sqrt{3} \) dans \( \mathbb{R} \) ?

a) \( x = -\frac{\pi}{12} + \frac{k\pi}{2} \)
b) \( x = -\frac{\pi}{6} + \frac{k\pi}{2} \)
c) \( x = -\frac{\pi}{4} + \frac{k\pi}{2} \)
d) \( x = -\frac{\pi}{3} + \frac{k\pi}{2} \)

8. Quelle est la solution de l’inéquation \( \cos x > \frac{1}{2} \) dans \( [-2\pi, 2\pi] \) ?

a) \( x \in \left]-\frac{\pi}{3}, \frac{\pi}{3}\right[ \cup \left]\frac{5\pi}{3}, \frac{7\pi}{3}\right[ \)
b) \( x \in \left]-\frac{\pi}{6}, \frac{\pi}{6}\right[ \cup \left]\frac{11\pi}{6}, \frac{13\pi}{6}\right[ \)
c) \( x \in \left]-\frac{\pi}{4}, \frac{\pi}{4}\right[ \cup \left]\frac{7\pi}{4}, \frac{9\pi}{4}\right[ \)
d) \( x \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[ \cup \left]\frac{3\pi}{2}, \frac{5\pi}{2}\right[ \)

9. Quelle est la solution de l’inéquation \( \sin x > -\frac{1}{2} \) dans \( [0, 2\pi] \) ?

a) \( x \in \left]0, \frac{7\pi}{6}\right[ \cup \left]\frac{11\pi}{6}, 2\pi\right[ \)
b) \( x \in \left]0, \frac{\pi}{6}\right[ \cup \left]\frac{5\pi}{6}, 2\pi\right[ \)
c) \( x \in \left]0, \frac{\pi}{4}\right[ \cup \left]\frac{3\pi}{4}, 2\pi\right[ \)
d) \( x \in \left]0, \frac{\pi}{3}\right[ \cup \left]\frac{2\pi}{3}, 2\pi\right[ \)

10. Quelle est la solution de l’inéquation \( \cos x < -\frac{\sqrt{2}}{2} \) dans \( [-\pi, \pi] \) ?

a) \( x \in \left]-\frac{3\pi}{4}, -\frac{\pi}{4}\right[ \)
b) \( x \in \left]-\frac{5\pi}{6}, -\frac{\pi}{6}\right[ \)
c) \( x \in \left]-\frac{2\pi}{3}, -\frac{\pi}{3}\right[ \)
d) \( x \in \left]-\frac{\pi}{2}, -\frac{\pi}{4}\right[ \)

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