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[parent] proof of cosines law (Proof)

Let $a$ , $b$ , $c$ be the sides of a triangle and $\alpha$ , $\beta$ , $\gamma$ its angles, respectively. By the projection formula, one may write the equalities

\begin{align*}\begin{cases}a = b\cos\gamma+c\cos\beta\\ b = c\cos\alpha+a\cos\gamma\\ c = a\cos\beta+b\cos\alpha. \end{cases}\end{align*}    

Multiplying the equalities by $a$ , $-b$ and $-c$ , respectively, they read
\begin{align*}\begin{cases}a^2 \;=\; ab\cos\gamma+ca\cos\beta\\ -b^2 = -bc\cos\alpha-ab\cos\gamma\\ -c^2 = -ca\cos\beta -bc\cos\alpha. \end{cases}\end{align*}    

Addition of these yields the sum equation $$a^2-b^2-c^2 = -2bc\cos\alpha,$$ i.e. $$a^2 \;=\; b^2+c^2-2bc\cos\alpha,$$ which is the cosines law.



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Cross-references: cosines law, equation, sum, addition, equalities, projection formula, angles, triangle, sides

This is version 1 of proof of cosines law, born on 2008-10-01.
Object id is 11116, canonical name is ProofOfCosinesLaw.
Accessed 416 times total.

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AMS MSC: 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
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