[tex]\cos\theta=1+3\cos\theta\implies 0=1+2\cos\theta\implies \cos\theta=-\dfrac12[/tex]
On the unit cirlce, the cosine of an angle is negative whenever the angle falls in the interval [tex]\dfrac\pi2<x<\dfrac{3\pi}2[/tex].
If you recall your special triangles, this cosine occurs for angles of [tex]x=\dfrac{2\pi}3[/tex] and [tex]x=\dfrac{4\pi}3[/tex].
More generally, this occurs for [tex]x=\dfrac{2\pi}3+2n\pi[/tex] and [tex]x=\dfrac{4\pi}3+2n\pi[/tex], where [tex]n[/tex] is an integer.
