Exercice : Somme 2
Question
Écrire le plus simplement possible. Expliquer brièvement mais soigneusement
1.\( sin(\frac{\pi}{9})- sin(\frac{7\pi}{9})+sin(\frac{2\pi}{9})+sin(\frac{10\pi}{9})\)
Solution
\(sin(\frac{\pi}{9})- sin(\frac{7\pi}{9})+sin(\frac{2\pi}{9})+sin(\frac{10\pi}{9})\)
\(=2sin(\frac{\frac{\pi}{9} –\frac{7\pi}{9}}{2})cos(\frac{\frac{\pi}{9}+\frac{7\pi}{9}}{2})+2sin(\frac{\frac{2\pi}{9} +\frac{10\pi}{9}}{2})cos(\frac{\frac{2\pi}{9}-\frac{10\pi}{9}}{2})\)
\(=2sin(\frac{\frac{-6\pi}{9}}{2})cos(\frac{\frac{8\pi}{9}}{2})+2sin(\frac{\frac{12\pi}{9}}{2})cos(\frac{\frac{-8\pi}{9}}{2})\)
\(=2(-sin(\frac{\pi}{3})cos(\frac{4\pi}{9})+2sin(\frac{2\pi}{3})cos(\frac{-4\pi}{9})\)
\(=-2sin(\frac{\pi}{3}cos(\frac{4\pi}{9})+2sin(\frac{2\pi}{3})cos(\frac{4\pi}{9})\)
car \(cos(-x)=cos(x)\)
\(=-2sin(\frac{\pi}{3}cos(\frac{4\pi}{9})+2sin(\frac{\pi}{3})cos(\frac{4\pi}{9})\)
car \(sin(\pi-x)=sin(x)\)
\(=-2sin(\frac{\pi}{3}cos(\frac{4\pi}{9})+2sin(\frac{\pi}{3})cos(\frac{4\pi}{9})\)
=0
Question
\(2. sin(\frac{\pi}{2}-x)+cos(x+2\pi)+sin(x+3\pi)+sin(x+2\pi)\)
Solution
\(sin(\frac{\pi}{2}-x)+cos(x+2\pi)+sin(x+3\pi)+sin(x+2\pi)\)
=\(cos(x)+cos(x)-sin(x)+sin(x)\)
=\(2cos(x)\)
Question
\(3. cos^2(x)-1+2sin^2x\)
Solution
\(cos^2(x)-1+2sin^2x\)
\(=1-sin^2x-1+2sin^2x\)
\(=sin^2x\)
Question
\(4. sin 0 + sin \frac{\pi}{3}+sin \frac{2\pi}{3}+sin\pi+sin \frac{4\pi}{3}+sin \frac{5\pi}{3}\)
Solution
\(sin 0 + sin \frac{\pi}{3}+sin \frac{2\pi}{3}+sin\pi+sin \frac{4\pi}{3}+sin \frac{5\pi}{3}\)
\(=0+\frac{1}{2}+\frac{1}{2}+0-\frac{1}{2}-\frac{1}{2}\)
\(=0\)