Question

If cos theta =-2/3, which of the following are possible

Answer 1

Well, the cos(∅) is negative in the second and third quadrants.  If you are solving for theta, you would use the inverse of the cos or arccos
$cos(x) = ( \frac{-2}{3} )$ take the inverse of both sides to get:
x = arccos(-2/3) now evaluate the right
x = 131.8103149 degrees
to find your second solution, subtract your reference angle from 360 degrees.
360 - 131.8103149 = 228.1896851 degrees

Now the period of cos is 2π or 360 degrees. So if you want to consider all possible solutions, you would need to add/subtract 360n to both solutions above.. 

Not sure if this is what you're looking for, but thought I would leave it here for you just in case... As a side note, you could do this problem in radian measurement as well.