Inverse Functions: Arccosine and SSS (Part 20)

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

Solve \triangle ABC if latex a = 16$, b = 20, and c = 40.

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

c^2 = a^2 + b^2 - 2 a b \cos \gamma

1600 = 256 + 400 - 640 \cos \gamma

944 =-640 \cos \gamma

-1.475 = \cos \gamma

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 a, b, and c are all positive)

c > a + b \Longleftrightarrow c^2 > (a+b)^2

\Longleftrightarrow a^2 + b^2 - 2 a b \cos \gamma > a^2 + 2 a b + b^2

\Longleftrightarrow -2 a b \cos \gamma > 2 a b

\Longleftrightarrow \cos \gamma < -1

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

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