calculator issue: radians or degrees for inverse trig It's a simple question but I am a little confused.  The value of $cos^{-1} (-0.5)$ ,  is it   2.0943 or 120  ? 

The principal value of $\cos^{-1}(-0.5)$ is $120^{\circ}$ or $\frac{2 \pi}{3} \approx 2.0943$ radians.  Take your pick!
A couple of key points here.
First, it helps to have a feeling for what range of values to expect when you do an operation.  If I'm taking $\cos^{-1}x$ I expect to get an angle.  If the angle is measured in degrees, reasonable values are $0$ to $180$.  If the angle is measure in radians, reasonable values are $0$ to $\pi \approx 3.14159265$.
For angles in general, $-180$ to $360$ would be resonable for degree measures, and $-3.14159265$ to $6.2831853$ or $-\pi$ to $2\pi$ would be reasonable for radian measures.
Second, though, any intuition should be verified by knowing what you're looking at.  You should know whether you need to input angles in degrees or radians, and you should know whether the angles you get back are expressed in degrees or in radians.
Because if you don't know, your intuition may not be enough.  What if you're calculating $\cos(\pi / 100)$?  Here, $\pi/100$ is an angle in radians.  You enter $0.031415926535$ and push the $\cos$ key.  If you're in "radians" mode, the answer you get back is $\approx 0.999506560$, but if you're instead in "degrees" mode, the answer you get back is $\approx 0.999999849$, which is substantially a different answer, but not one that you could easily tell was wrong just by looking at it.  "The angle is close to zero, $\cos$ should be close to one" wouldn't have been enough intuition to catch the error. 
