Question

Expression for all solutions to the equation cos(theta) = 1 + 3cos(theta)

Answer 1

$\cos\theta=1+3\cos\theta\implies 0=1+2\cos\theta\implies \cos\theta=-\dfrac12$

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.