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 answer depends on whether you are working in degrees or radians. It's just a case of units.
When we think about length, we have $1 \, \text{km} = 1,000\,\text{m}$. 
When we talk about velocity we have $1 \, \text{km/h} = \tfrac{8}{15}\text{m/s}$.
When we talk about angles we have $180^{\circ} = \pi^c$.
However, when we are doing calculus: we must always use radians. 
The usual definition of cosine is in terms of the ratio of the adjacent and the hypotenuse of a right-angled triangle. In this case, degrees or radians are fine. There is another "definition" of the cosine function which comes from the Taylor Series:
$$\cos x \sim 1 - \tfrac{1}{3!}x^3 + \tfrac{1}{5!}x^5 - \tfrac{1}{7!}x^7 + \cdots $$
This definition depends on differentiation, i.e. on using calculus. Hence: you must use radians.
