Correct up to here.

You need to take the reciprocal of both sides (you only did the right hand side)sec^-1 (-3/2) = -2x + 180

This is where I'm running into trouble. I know that sec = 1/cos, so I rewrote this line as:

1/(cos^-1 (-3/2) ) = -2x + 180

But I'm not able to solve for cos^-1 (-3/2) so now I'm at a bit of a loss.