This investigation of cosine triples was instigated by a chance observation. For years I have used triangles with integral sides for the purpose of quizzing students on the law of cosines. The law of cosines states that the square of the length of any side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides and the cosine of the angle between them. For example, given the triangle with sides 6 and 7, and included angle 20 degrees, the student is expected to find the remaining side using the law of cosines (fig. 1).