triginometry

Determine the angle when the adjacent measurement is 4.870 in and the opposite angle is 3.900 in

Your question doesn't make sense.
I could guess what you were talking about but there would be more than one 'guess' involved.

So i think you're trying to say, from a given angle (lets call it X) the adjacent side is 4.870 inches and the opposite side is 3.900 inches. What is the angle?
So if that is indeed your question this is a TanX formula. TanX = opposite/adjacent. So TanX = 4.870/3.900 or about 1.249.
Now X does not equal 1.249. TanX = 1.249. Then do the inverse Tan (or Tan^-1) of 1.249 on a trig calculator (or if you're retro, use the back of an old trig textbook) and you get 51.3 degrees (or .896 radians)