Chapitre X Produit scalaire

\(\frac{\pi}{6}+\frac{\pi}{4}=\frac{2\pi}{12}+\frac{3\pi}{12}=\frac{5\pi}{12}\)

\(cos(\frac{5\pi}{12})=cos(\frac{\pi}{6}+\frac{\pi}{4})=cos(\frac{\pi}{6})cos(\frac{\pi}{4})-sin(\frac{\pi}{6})sin(\frac{\pi}{4})=\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}-\frac{1}{2} \times \frac{\sqrt{2}}{2}\)

\(\iff cos(\frac{5\pi}{12})=\frac{\sqrt{3}}{4}-\frac{\sqrt{2}}{4}=\frac{\sqrt{3}-\sqrt{2}}{4}\)

\(sin(\frac{5\pi}{12})=sin(\frac{\pi}{6}+\frac{\pi}{4})=sin(\frac{\pi}{6})cos(\frac{\pi}{4})+cos(\frac{\pi}{6})sin(\frac{\pi}{4})=\frac{1}{2} \times \frac{\sqrt{2}}{2}+\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}\)

\(\iff sin(\frac{5\pi}{12})=\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}\)

\(\iff sin(\frac{5\pi}{12})=\frac{\sqrt{2}+\sqrt{6}}{4}\)

\(\frac{2\pi}{3}+\frac{\pi}{4}=\frac{8\pi}{12}+\frac{3\pi}{12}=\frac{11\pi}{12}\)

\(cos(\frac{11\pi}{12})=cos(\frac{2\pi}{3}+\frac{\pi}{4})=cos(\frac{2\pi}{3})cos(\frac{\pi}{4})-sin(\frac{2\pi}{3})sin(\frac{\pi}{4})=\frac{-1}{2} \times \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}\)

\(\iff cos(\frac{11\pi}{12})=\frac{-1}{2} \times \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}=\frac{-\sqrt{2}}{4}-\frac{\sqrt{6}}{4}\)

\(\iff cos(\frac{11\pi}{12})=-(\frac{\sqrt{2}+\sqrt{6}}{4})\)

\(sin(\frac{11\pi}{12})=sin(\frac{2\pi}{3}+\frac{\pi}{4})=sin(\frac{2\pi}{3})cos(\frac{\pi}{4})+cos(\frac{2\pi}{3})sin(\frac{\pi}{4})=\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}+\frac{-1}{2} \times \frac{\sqrt{2}}{2}\)

\(\iff sin(\frac{11\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)

\(\iff sin(\frac{11\pi}{12})=\frac{\sqrt{6}-\sqrt{2}}{4}\)