Réponses

  • Utilisateur Brainly
2013-06-12T19:11:37+02:00

Il faut utiliser les formules d'Euler :

sin(5x)=Im [cos(5x)+isin(5x)]

            = Im[e^(5ix)]

            =Im[(e^(ix))^5]

            =Im((cos(x)+isin(x))^5]

 

or(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5

donc on obtient :

(cos(x)+isin(x))^5=(cos x)^5+5(cos x)^4(i sin x)+10(cos x)^3(i sin x)^2+

                               +10(cos x)^2(i sin x)^3+5(cos x)(i sin x)^4+(i sin x)^5

                               =(cos x)^5+5(cos x)^4(sin x) i -10(cos x)^3(sin x)^2+

                               -10 i (cos x)^2(sin x)^3+5(cos x)(sin x)^4-i (sin x)^5

 

donc sin(5x)=5(cos x)^4(sin x) -10 (cos x)^2(sin x)^3-(sin x)^5