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Law of Cosines, a la Shmoop. Anthony, Bernardo and Chuckles, three members
of the mafia…
…were all contracted to whack one another.
They all ended up in the same place at the same time, and next thing they knew, they
were trapped in a standoff. There is 45 feet between Anthony and Bernardo…
…74 feet between Bernardo and Chuckles…
…and 61 feet between Chuckles and Anthony.
What are the angles between each? If we picture the scenario as a triangle,
the side are 45, 74 and 61.
Given three sides of the triangle, we’ll need to figure out the angles with the law
of cosines. The Law of Cosines tells us that a squared
equals b squared plus c squared minus 2 b times c times cosine of the angle A.
Before we plug in all the values we know, let’s isolate what we’re solving for:
the angle A. Subtract a squared and add 2bc cosine A to
both sides of the equation to get 2bc cosine A equals b squared plus c squared minus a
squared. Then divide both sides by 2bc to get cos A
equals b squared plus c squared minus a squared all over 2bc.
To isolate the angle, we can take the cosine inverse of both sides, and when we do, we
end up with the angle A equals the cosine inverse of b squared plus c squared minus
a squared all over 2bc. Now let’s plug in b as 61, c as 45, and
a as 74. Plugging it into our calculators, we see that
angle A equals around 87.2 degrees. To find angle B, we can use the Law of Cosines
again…and we find that b equals around 55.4 degrees.
We know that all the angles in a triangle total 180 degrees…
…so we can just take 180 degrees minus 87.2 minus 55.4 to find angle C…
…which gives us 37.4 degrees. However, it’s going to take more than a
basic understanding of cosines to get Anthony, Bernardo and Chuckles out of their pickle.
They might be here a while. Good thing somebody