Chapitre X Produit scalaire

\(sin \frac{\pi}{8} – sin \frac{3\pi}{8} + sin \frac{5\pi}{8} – sin\frac{7\pi}{8}\)

\(=2sin(\frac{\frac{\pi}{8} –\frac{3\pi}{8}}{2})cos(\frac{\frac{\pi}{8}+\frac{3\pi}{8}}{2})+2sin(\frac{\frac{5\pi}{8} –\frac{7\pi}{8}}{2})cos(\frac{\frac{5\pi}{8}+\frac{7\pi}{8}}{2})\)

\(=2sin(\frac{\frac{-2\pi}{8}}{2})cos(\frac{\frac{4\pi}{8}}{2})+2sin(\frac{\frac{-2\pi}{8}}{2})cos(\frac{\frac{12\pi}{8}}{2})\)

\(=2sin(\frac{-\pi}{8})cos(\frac{2\pi}{8})+2sin(\frac{-\pi}{8})cos(\frac{6\pi}{8})\)

\(=2sin(\frac{-\pi}{8})cos(\frac{\pi}{4})+2sin(\frac{-\pi}{8})cos(\frac{3\pi}{4})\)

\(=2sin(\frac{-\pi}{8}) \times \frac{\sqrt{2}}{2}-2sin(\frac{\pi}{8}) \times (-\frac{\sqrt{2}}{2})\)

\(=-2sin(\frac{\pi}{8}) \times \frac{\sqrt{2}}{2}+2sin(\frac{\pi}{8}) \times\frac{\sqrt{2}}{2}=0\)