Cos a = 0.83 and cos b = 0.55. what is sin a + sin b?

If you were looking for sin (A+B), this would be a more familiar problem.

There's an identity that'd apply here:

sin (A+B)=sin A cos B + cos A sin B = sin A (0.55) + (0.83) sin B.  But we don't have the value of sin A or that of sin B.

So, I will make the assumption that angles A and B pertain to unit circles, that is, that the hypotenuse of these angles are both 1.  In that case it takes just the application of the Pythagorean Theorem to find sin A and sin B.

If cos A = 0.83, then (cos A)^2 + (sin A)^2 = 1^2 = 1, so that 
(sin A)^2 = 1 - (cos A)^2, or    sin A = plus or minus sqrt(1 - (cos A)^2).

We'd find sin B in the same way.  Then, we could write out sin A + sin B.

Forgive my asking, but I'd like for you to double-check to ensure that you have copied and shared all of the given problem.  Is there any mention of "unit circle?"