To solve the equation [tex]\sqrt{3}\cos(x) + \sin(x) = 0[/tex], we can rewrite it as:

[tex]\sqrt{3}\cos(x) = -\sin(x)[/tex]

Now, we can square both sides to eliminate the square root:

tex^2 = (-\sin(x))^2[/tex]

[tex]3\cos^2(x) = \sin^2(x)[/tex]

Using the trigonometric identity [tex]\sin^2(x) + \cos^2(x) = 1[/tex], we have:

[tex]3\cos^2(x) + \cos^2(x) = 1[/tex]

[tex]4\cos^2(x) = 1[/tex]

[tex]\cos^2(x) = \frac{1}{4}[/tex]

[tex]\cos(x) = \pm \frac{1}{2}[/tex]

So, [tex]x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}, \frac{11\pi}{3}[/tex]