Answer
$1.82+2k\pi$, $4.46+2k\pi$
Work Step by Step
$\begin{align}
4\cos\theta+1&=0\\
4\cos\theta&=-1\\
\cos\theta&=-\frac{1}{4}
\end{align}$
Since $\cos \theta$ is negative in the second and third quadrants, the only values of $\theta$ between $0$ and $2\pi$ that satisfy the equation are $\cos^{-1}(-\frac{1}{4})\approx 1.82$ and $2\pi-\cos^{-1}(-\frac{1}{4})\approx 4.46$.
Since $\cos \theta$ is a periodic function with period $2\pi$, we can add or subtract any multiple of $2\pi$ from $1.82$ or $4.46$ and the equation would still be satisfied.
So the answers are $1.82+2k\pi$ and $4.46+2k\pi$ for any integer $k$.
