how to relate the law of cosine to the real world?

Answer 1

It is used frequently in surveying when the formulas for right angle triangles are not applicable.

Answer 2

Well, the law of cosines solves for sides and angles other than right triangles, so how about every single other triangle in the world. Like all the bits that engineered the Eiffel tower or the Golden Gate Bridge or the Hancock Building in Chicago.

Also, in non-Euclidean spaces, the right triangle may not be solveable using the simple Pythagorean theorem. In a curved space like the surface of the Earth, you need the law of cosines to solve for Great Circles like airplane flights and the paths of sub-oceanic cables.