Question

cosx + cos2x + cos3x + 1 = 0

Answer 1

Lời giải:
\(\cos x+\cos 2x+\cos 3x+1=0\)

\(\Leftrightarrow (\cos 2x+1)+\cos x+\cos 3x=0\)

\(\Leftrightarrow (2\cos ^2x-1+1)+2.\cos \frac{3x+x}{2}\cos \frac{3x-x}{2}=0\)

\(\Leftrightarrow 2\cos ^2x+2\cos 2x\cos x=0\)

\(\Leftrightarrow \cos x(\cos x+\cos 2x)=0\)

TH1: $\cos x=0$

$\Rightarrow x=\frac{\pi}{2}+k\pi$ (với $k$ nguyên)

TH2: $\cos x+\cos 2x=0$

$\Leftrightarrow \cos x+2\cos ^2x-1=0$

$\Leftrightarrow (2\cos x-1)(\cos x+1)=0$

$\Rightarrow \cos x=\frac{1}{2}$ hoặc $\cos x=-1$

$\Rightarrow x=\pm \frac{\pi}{3}+2k\pi$ với $k$ nguyên

Hoặc $x=(2k+1)\pi$ với $k$ nguyên