Answer
The corresponding values of $\theta$ are $$\{135^\circ,180^\circ,225^\circ,270^\circ\}$$
Work Step by Step
1) The question asks for solutions over the interval $[0^\circ, 360^\circ)$, meaning that
$$\theta\in[0^\circ,360^\circ)$$
or $$0^\circ\le\theta\lt360^\circ$$
2) Now from also the question, your work leads to $$\frac{1}{3}\theta=45^\circ,60^\circ,75^\circ,90^\circ$$
To find corresponding values of $\theta$ from $\frac{1}{3}\theta$, we multiply both sides by $3$
$$\theta=135^\circ,180^\circ,225^\circ,270^\circ$$
3) Finally, it is crucial to compare the values of $\theta$ in 2) with the range found in 1).
We see that all four values of $\theta=135^\circ,180^\circ,225^\circ,270^\circ$ fit in the range of $\theta\in[0^\circ,360^\circ)$. None would be eliminated as a result.
The corresponding values of $\theta$ are $$\{135^\circ,180^\circ,225^\circ,270^\circ\}$$
