I did the following steps to get my answer of $-\pi/4$.
Subtract as many $2\pi$ as possible from $77\pi/4$, which gives $77\pi/4-18\pi = 5\pi/4$.
This leaves the angle in the 3rd quadrant. I believe that the domain for this function is $[-\pi/2,\pi/2]$. So, subtract $\pi$ from $5\pi/4$, which gives $-\pi/4$.
However, I am not sure that this is the right answer.
Should I get something different? If so, how?
All help is appreciated.
Step 1 is good, but you've used the wrong range for arccosine. It is between $0$ and $\pi$. So you get your $5\pi/4$, in the third quadrant; $\pi/4$ beyond $\pi$.
What's an angle in the interval $[0, \pi]$ with the same cosine? Well, thinking geometrically, a reflection across the $x$-axis preserves cosine. So since $5\pi/4 = \pi + \pi/4$, the angle you're looking for must be $\pi - \pi/4 = 3\pi/4$.
Note that cosine isn't one-to-one on the interval $[-\pi/2, \pi/2]$ -- for example, $\cos(\pi/4) = \cos(-\pi/4)$ -- so this wouldn't make a very good range for its inverse! It is the range for the inverse of sine and tangent, however, so it's a common mistake.