1. Calculer \( \sin\left(\frac{53\pi}{6}\right) \) :
\( \frac{1}{2} \)2. Calculer \( \cos\left(-\frac{29\pi}{6}\right) \) :
\( \frac{1}{2} \)3. Sachant que \((u, v) \equiv -\frac{\pi}{7} \ [2\pi]\) et \((u, w) \equiv -\frac{\pi}{4} \ [2\pi]\), quelle est la mesure principale de \((v, w)\) ?
4. Que vaut l’expression \(A = \cos(0) + \cos\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2}\right) + \cos\left(\frac{3\pi}{4}\right) + \cos(\pi)\) ?
5. Sachant que \(\cos x = -\frac{3}{4}\) et \(-\pi < x < 0\), que vaut \(\sin x\) ?
6. Calculer \( \tan\left(-\frac{16\pi}{3}\right) \) :
\( \sqrt{3} \)7. Calculer \( \sin\left(-\frac{19\pi}{4}\right) \) :
\( \frac{\sqrt{2}}{2} \)8. Calculer \( \cos\left(\frac{37\pi}{2}\right) \) :
09. Calculer \( \tan(2025\pi) \) :
010. Montrer que \( 1 + \tan^2 x = \frac{1}{\cos^2 x} \) :
Vrai11. Si \( \sin x = -\frac{4}{5} \) et \( -\frac{\pi}{2} < x < \frac{\pi}{2} \), calculer \( \cos x \) :
\( -\frac{3}{5} \)12. Si \( \tan(x) = \frac{1}{3} \) et \( \frac{\pi}{2} < x < \pi \), calculer \( \cos x \) :
\( -\frac{3}{\sqrt{10}} \)13. Si \( \tan(x) = \frac{1}{3} \) et \( \frac{\pi}{2} < x < \pi \), calculer \( \sin x \) :
\( -\frac{1}{\sqrt{10}} \)14. Simplifier \( A(x) = \sin(-x) – \cos(-x) \) :
\( -\sin x – \cos x \)15. Simplifier \( B(x) = \sin(\pi + x) + \cos(\pi + x) \) :
\( -\sin x – \cos x \)16. Simplifier \( C(x) = \sin(3\pi + x) + \cos(2\pi + x) \) :
\( -\sin x + \cos x \)17. Simplifier \( D(x) = \cos(\pi + x) + \sin(-x) + \sin(x – 4\pi) \) :
\( -\cos x – \sin x \)18. Simplifier \( A(x) = \sin\left(\frac{\pi}{2} – x\right) + \cos(3\pi – x) + \sin\left(x – \frac{\pi}{2}\right) \) :
\( \cos x – \sin x \)19. Sachant que \( \cos\left(\frac{9\pi}{5}\right) = \frac{1 + \sqrt{5}}{4} \), calculer \( \sin\left(\frac{9\pi}{5}\right) \)
\( -\frac{\sqrt{10 – 2\sqrt{5}}}{4} \)20. En déduire \( \cos\left(\frac{\pi}{5}\right) \) :
\( \frac{1 + \sqrt{5}}{4} \)