Answer
$$A= 30+ n360^{\circ},\ \ \text{or}\ \ 270^{\circ}+n360^{\circ} $$
where $n $ is integer
Work Step by Step
Given
$$ \cos (A+30^{\circ}) =\left( \frac{1}{2}\right)$$
Since
\begin{align*}
A+30^{\circ}&=\cos^{-1}\left( \frac{1}{2}\right)\\
A+30^{\circ}&=60^{\circ},\ \ \text{or}\ \ 300^{\circ}\\
A&= 30^{\circ},\ \ \text{or}\ \ 270^{\circ}
\end{align*}
Since the period of $\cos$ function is $360^{\circ}$ , then the general solution is
$$A= 30+ n360^{\circ},\ \ \text{or}\ \ 270^{\circ}+n360^{\circ} $$
where $n $ is integer