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