Question

Escreva sin 3x e cos 3x em termos de sen x e cos x, respectivamente

Answer 1

Olá, Misinho.


$\sin3x=\sin (2x + x) = \sin2x\cos x + \sin x\cos2x=\\\\ =2\sin x\cos x\cos x+\sin x(\cos^2x-\sin^2x)=\\\\ =2\sin x\cos^2x+\sin x\cos^2x-\sin^3x=\\\\ =3\sin x\cos^2x-\sin^3x$


$\cos3x=\cos(2x+x)=\cos2x\cos x-\sin2x\sin x=\\\\ =(\cos^2x-\sin^2x)\cos x-2\sin x\cos x\sin x=\\\\ =\cos^3x-\sin^2x\cos x-2\sin^2x\cos x=\\\\ =\cos^3x-3\sin^2x\cos x$