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Law of Cosines, a la Shmoop. Anthony, Bernardo and Chuckles, three members

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of the mafia…

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…were all contracted to whack one another.

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They all ended up in the same place at the same time, and next thing they knew, they

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were trapped in a standoff. There is 45 feet between Anthony and Bernardo…

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…74 feet between Bernardo and Chuckles…

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…and 61 feet between Chuckles and Anthony.

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What are the angles between each? If we picture the scenario as a triangle,

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the side are 45, 74 and 61.

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Given three sides of the triangle, we’ll need to figure out the angles with the law

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of cosines. The Law of Cosines tells us that a squared

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equals b squared plus c squared minus 2 b times c times cosine of the angle A.

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Before we plug in all the values we know, let’s isolate what we’re solving for:

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the angle A. Subtract a squared and add 2bc cosine A to

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both sides of the equation to get 2bc cosine A equals b squared plus c squared minus a

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squared. Then divide both sides by 2bc to get cos A

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equals b squared plus c squared minus a squared all over 2bc.

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To isolate the angle, we can take the cosine inverse of both sides, and when we do, we

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end up with the angle A equals the cosine inverse of b squared plus c squared minus

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a squared all over 2bc. Now let’s plug in b as 61, c as 45, and

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a as 74. Plugging it into our calculators, we see that

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angle A equals around 87.2 degrees. To find angle B, we can use the Law of Cosines

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again…and we find that b equals around 55.4 degrees.

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We know that all the angles in a triangle total 180 degrees…

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…so we can just take 180 degrees minus 87.2 minus 55.4 to find angle C…

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…which gives us 37.4 degrees. However, it’s going to take more than a

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basic understanding of cosines to get Anthony, Bernardo and Chuckles out of their pickle.

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They might be here a while. Good thing somebody

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ordered pizza.

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