Question

Calculați cosx+ cos2x+ cos3x=0

Answer 1

Rezolvarea se afla in fotografia de mai jos. 

Answer 2

cosx+ cos2x+ cos3x=0

$\it cos3x+cosx = 2cos\dfrac{3x+x}{2}cos\dfrac{3x-x}{2} =2cos\dfrac{4x}{2}cos\dfrac{2x}{2} =2cos2x \cos x$

Ecuația devine:

[tex]\it cos2x+2cos2x cosx=0 \Rightarrow cos2x(1+2cosx) =0 \Rightarrow\begin{cases} \it cos2x=0 \\\;\\ \it 1+2cosx=0 \end{cases}[/tex]


$\it cos2x=0 \Rightarrow 2x\in \left\{\dfrac{\pi}{2} +2k\pi|\ k\in\mathbb{Z} \right\} \Rightarrow x\in \left\{\dfrac{\pi}{4} +2k\pi|\ k\in\mathbb{Z} \right\}$

[tex]\it 1+2cosx = 0 \Rightarrow cosx=-\dfrac{1}{2} \\\;\\ \\\;\\ x\in \left\{\dfrac{2\pi}{3} +2k\pi|\ k\in\mathbb{Z} \right\} \cup \left\{\dfrac{7\pi}{6} +2k\pi|\ k\in\mathbb{Z} \right\} [/tex]