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=-\frac12`

On the unit cirlce, the cosine of an angle is negative whenever the angle falls in the interval
`\frac\pi2<x<\frac{3\pi}2`.

If you recall your special triangles, this cosine occurs for angles of
`x=\frac{2\pi}3` and
`x=\frac{4\pi}3`.

More generally, this occurs for
`x=\frac{2\pi}3+2n\pi` and
`x=\frac{4\pi}3+2n\pi`, where
`n` is an integer.