Answer
$$A= \frac{7\pi}{9}+2 n\pi,\ \ \text{or}\ \ \frac{13\pi}{9}+2 n\pi$$
where $n $ is integer
Work Step by Step
Given
$$ \cos \left(A-\frac{\pi}{9}\right) =\left( \frac{-1}{2}\right)$$
Since
\begin{align*}
A-\frac{\pi}{9}&=\cos^{-1}\left( \frac{-1}{2}\right)\\
A-\frac{\pi}{9}&=\frac{2\pi}{3},\ \ \text{or}\ \ \frac{4\pi}{3}\\
A&=\frac{7\pi}{9},\ \ \text{or}\ \ \frac{13\pi}{9}
\end{align*}
Since the period of $\cos$ function is $2\pi $ , then the general solution is
$$A= \frac{7\pi}{9}+2 n\pi,\ \ \text{or}\ \ \frac{13\pi}{9}+2 n\pi$$
where $n $ is integer