Question

Sa se transforme in produs

cos2x + cos3x + cos4x​

Answer 1

Salut,

Notăm cu E expresia din enunț.

Știm formula de calcul prescurtat:

$cosa+cosb=2\cdot cos\dfrac{a-b}2\cdot cos\dfrac{a+b}2.$

Expresia din enunț devine succesiv:

$E=cos(4x)+cos(2x)+cos(3x)=2\cdot cos\dfrac{4x-2x}2\cdot cos\dfrac{4x+2x}2+cos(3x)=\\\\=2\cdot cosx\cdot cos(3x)+cos(3x)=(2\cdot cosx+1)\cdot cos(3x)=2\cdot\left(cosx+\dfrac{1}2\right)\cdot cos(3x)=\\\\=2\cdot\left(cosx+cos\dfrac{\pi}3\right)\cdot cos(3x)=2\cdot 2\cdot cos\dfrac{x-\dfrac{\pi}3}2\cdot cos\dfrac{x+\dfrac{\pi}3}2\cdot cos(3x)=\\\\=4\cdot cos\dfrac{3x-\pi}6\cdot cos\dfrac{3x+\pi}6\cdot cos(3x).$

Ai înțeles rezolvarea ?

Green eyes.