Exercice
Calculer les valeurs suivantes (la réponse non justifiée n'a aucune valeur)
Question
1.\(sin\frac{3\pi}{8}+sin\frac{5\pi}{8}+sin\frac{11\pi}{8}+sin\frac{13\pi}{8}\)
Solution
\(sin\frac{3\pi}{8}+sin\frac{5\pi}{8}+sin\frac{11\pi}{8}+sin\frac{13\pi}{8}\)
\(=2sin(\frac{\frac{3\pi}{8}+\frac{5\pi}{8}}{2})cos(\frac{\frac{3\pi}{8}-\frac{5\pi}{8}}{2})+\)
\(2sin(\frac{\frac{11\pi}{8}+\frac{13\pi}{8}}{2})cos(\frac{\frac{11\pi}{8}-\frac{13\pi}{8}}{2})\)
\(=2sin(\frac{\pi}{2})cos(\frac{\frac{-2\pi}{8}}{2})+\)
\(2sin(\frac{\frac{11\pi+13\pi}{8}}{2})cos(\frac{\frac{-2\pi}{8}}{2})\)
\(=2cos(\frac{-\pi}{8})+2sin(\frac{\frac{24\pi}{8}}{2})cos(\frac{-\pi}{8})\)
\(=2cos(\frac{-\pi}{8})+2sin(\frac{3\pi}{2})cos(\frac{-\pi}{8})\)
\(=2cos(\frac{-\pi}{8})-2cos(\frac{-\pi}{8})\)
\(=2cos(\frac{\pi}{8})-2cos(\frac{\pi}{8})\)
\(=0\)
Question
\(cos\frac{\pi}{10}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}+cos\frac{9\pi}{10}\)
Solution
\(cos\frac{\pi}{10}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}+cos\frac{9\pi}{10}\)
\(=cos\frac{\pi}{10}+cos\frac{9\pi}{10}+cos\frac{2\pi}{5}+cos\frac{3\pi}{5}\)
\(=2cos(\frac{\frac{\pi}{10}+\frac{9\pi}{10}}{2})cos(\frac{\frac{\pi}{10}-\frac{9\pi}{10}}{2})+2cos(\frac{\frac{2\pi}{5}+\frac{3\pi}{5}}{2})cos(\frac{\frac{2\pi}{5}-\frac{3\pi}{5}}{2})cos(\frac{\frac{2\pi}{5}+\frac{3\pi}{5}}{2})\)
\(=2cos(\frac{\pi}{2})cos(-\frac{8\pi}{20})+2cos(\frac{\pi}{2})cos(-\frac{\pi}{10})cos(\frac{\pi}{2})\)
=0