[tex]3\tan3\theta-1=0[/tex]
[tex]3\tan3\theta=1[/tex]
[tex]\tan3\theta=\dfrac13[/tex]
Recall that the tangent function has a period of [tex]\pi[/tex] so that
[tex]3\theta=\tan^{-1}\dfrac13+k\pi[/tex]
for any integer [tex]k[/tex]. Then
[tex]\theta=\dfrac13\tan^{-1}\dfrac13+\dfrac{k\pi}3[/tex]
We get 6 solutions in the interval [0, 2π) for [tex]0\le k\le5[/tex],
[tex]\theta\approx0.107[/tex]
[tex]\theta\approx1.154[/tex]
[tex]\theta\approx2.202[/tex]
[tex]\theta\approx3.249[/tex]
[tex]\theta\approx4.296[/tex]
[tex]\theta\approx5.343[/tex]
