Exercice : Somme 4 [Chapitre X Produit scalaire]

Exercice : Somme 4

Calculer en mettant sous la forme la plus simple possible:

1. \(A=cos⁡(x)×sin⁡(\frac{π}{2}+x)-cos(\frac{π}{2}+x)×sin⁡(x)\)

\(sin(a-b)=sin(a)cos(b)-cos(a)sin(b)\)

\(A=cos⁡(x)×sin⁡(\frac{π}{2}+x)-cos(\frac{π}{2}+x)×sin⁡(x)\)

\(\iff A=sin⁡(\frac{π}{2}+x)\times cos⁡(x)-cos(\frac{π}{2}+x)×sin⁡(x)\)

\(\iff A=sin⁡(\frac{π}{2}+x-x)=sin⁡(\frac{π}{2})=1\)

\(2.B=sin⁡(x+\frac{π}{2})+sin⁡(x+π)+sin⁡(x+\frac{3π}{2})+sin⁡(x+2012π)\)

\(B=cos(x)-sin⁡(x)-cos⁡(x)+sin⁡(x)=0\)

\(C=sin(\frac{π}{5})+sin(\frac{7π}{10})+sin(\frac{4π}{5})+sin(\frac{3π}{10})\)

\(C=sin(\frac{π}{5})+sin(\frac{7π}{10})+sin(\frac{4π}{5})+sin(\frac{3π}{10})\)

\(\frac{π}{2}+\frac{π}{5}=\frac{5π}{10}+\frac{2π}{10}=\frac{7π}{10}\)

\(\frac{π}{2}-\frac{π}{5}=\frac{5π}{10}-\frac{2π}{10}=\frac{3π}{10}\)

\(π-\frac{π}{5}=\frac{5π}{5}-\frac{π}{5}=\frac{4π}{5}\)

\(\iff C=sin(\frac{π}{5})+sin(\frac{π}{2}+\frac{π}{5})+sin(π-\frac{π}{5})+sin(\frac{π}{2}-\frac{π}{5})\)

\(\iff C=sin(\frac{π}{5})+cos(\frac{π}{5})+sin(\frac{π}{5})+cos(\frac{π}{5})\)

\(\iff C=2cos(\frac{π}{5})+2sin(\frac{π}{5})=2(cos(\frac{π}{5})+sin(\frac{π}{5})\)