Law of cosines
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The law of cosines is a formula that helps in triangulation when two or three side lengths of a triangle are known. The formula relates all three side lengths of a triangle to the cosine of a particular angle.
When to use it: SAS, SSS.
Let be oriented so that is at the origin, and is at the point.
is the distance from to .
Substituting the appropriate points into the distance formula gives us
Squaring the inner terms, we have
Square both sides for
Complete the triangle using the law of cosines.
To find the side length ,
Since , substitution gives us
Taking the square root of both sides gives us
Now we can orient the triangle differently to get get a new version of the law of cosines so we can find angle measure ,
Substituting in the appropriate side lengths gives us
Subtracting from both sides gives us
Dividing both sides by gives us
Using inverse trig, we know that
And we can find the last angle measure by subtracting the other two measures from