In ∆ABC, m∠A = 60º, m∠C = 30º, and AB = 6 inches. What is the length of side BC?

ABC is a 30 60 90 triangle

In such a triangle, the shortest side AB multiplied by the square root of 3 equals the length of the "medium" side CB.

6 * sq root (3) = 10.39

(Incidentally, the hypotenuse AC = short side*2 = 12)