The Law of Cosines also recognizes when the purported sides of a triangle are impossible.

Solve latex a = 16$, , and .

Hopefully students would recognize that , thus quickly demonstrating that the triangle is impossible. However, this also falls out of the Law of Cosines:

Since the cosine of an angle can’t be less than -1, we can conclude that this is impossible.

Stated another way, we have the implications (since , , and are all positive)

Since the last statement is impossible, so is the first one.