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Unformatted text preview: 2 + b2 − 2bc cos(α)
The remaining formulas given in Theorem 11.5 can be shown by simply reorienting the triangle
to place a diﬀerent vertex at the origin. We leave these details to the reader. What’s important
about a and α in the above proof is that (α, a) is an angle-side opposite pair and b and c are the
sides adjacent to α – the same can be said of any other angle-side opposite pair in the triangle.
Notice that the proof of the Law of Cosines relies on the distance formula which has its roots in the
Pythagorean Theorem. That being said, the Law of Cosines can be thought of as a generalization
of the Pythagorean Theorem. If we have a triangle in which γ = 90◦ , then cos(γ ) = cos (90◦ ) = 0
so we get the familiar relationship c2 = a2 + b2 . What this means is that in the larger mathematical
sense, the Law of Cosines and the Pythagorean Theorem amount to pretty much the same thing.2
Example 11.3.1. Solve the following triangles. Give exact answers and decimal approximations
(rounded to hundredths) and sketch the triangle.
1. β = 50◦...
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