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.

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*Posted by John Quintanilla on October 20, 2014*

https://meangreenmath.com/2014/10/20/inverse-functions-arccosine-and-sss-part-20/