Simple Law of Cosines Question

Huge suspension bridges are build with the earth's roundness in mind.  The two towers are plumb line straight up and down and yet, because of their colossal size, they are a bit further apart at their tops than they are at their base.  So, how can we calculate what this difference would be?

Here is the input data:

If we know the earth's radius; the location of the base of the towers above sea level; The distance (from the center of each base of the tower) between the bases; and the height of the tower, how would we calculate the distance from the tops of the towers.

I made this video to explain what I am talking about but I want to have the mathematical formula to predict the distance differences between the tops of the towers compared to the base:
h t t p s : / / w w w . y o u t u b e . c o m / w a t c h ? v = 8 N u N g a 3 B p n s

I have seen a similar question answered once using something called "the law of cosines" where, if you know an angle and the length of two vectors, you can calculate the distances between the two vectors?  I hope that helps and gives us a clue.