There is no rule that you NEED to reverse the sign, for example it would be quite correct to go.
Cosx=-0.8 Arccos-0.8=143. That gives you theta, the top right quadrant, and now we go to the "cos" quadrant, bottom right, for 360-theta, 217. 143 and 217 is the correct answer. However, sometimes (i can't think of when), it is easier to reverse the sign, Arccos0.8=36.9, reverse the quadrants we use, so now top left and bottom left, "sin" and "tan" respectively, 180+36.9=217, 180-36.9=143.
In essence, all you need to know is for Sin, t and 180-t. Cos, t and 360-t. Tan, t and 180+t. In some circumstance, you find it's easier to calculate arccos of the reverse sign, or are given the actual value in a non calc paper, you use the other two equations.