The Law of Cosines (Independent of the Pythagorean Theorem)

The Law of Cosines (interchangeably known as the Cosine Rule or Cosine Law) can be shown to be a consequence of the Pythagorean theorem of which it is a generalization. Euclid proved his variant of the Law of Cosines in two propositions: who proved the obtuse case as II.12 for obtuse angles and II.13 for acute one. He used the Pythagorean theorem in both cases. For this reason, I expressed interest whether it is possible to prove the Law of Cosines independent of the latter.

My inquiry was rejected on the grounds that the two - the Pythagorean theorem and the Law of Cosines - are either both hold (in Euclidean geometry) or both do not hold (in spherical or hyperbolic geometries), implying their dependencies on each other. However, John Molokach came up with a proof of the Law of Cosines (see below) which does not appear to rely on the Pythagorean theorem. How can this be explained?

There is no paradox here. The two statements - the Pythagorean theorem and the Law of Cosines - are indeed equivalent. In any given geometry, they are either both true or both false. However, both may stem (quite independently) from another - perhaps, but not necessarily, more fundamental - proposition.

Theorem

For a triangle with sides $a$,$b$, and $c$ and the angle $\gamma$ opposite the side $c$, one has

$c^{2} = a^{2} + b^{2} - 2ab\cdot\cos\gamma.$

Proof
John Molokach
2 March, 2011

In any triangle with sides $a,$ $b,$ $c$ and opposite angles $\alpha ,$ $\beta ,$ $\gamma ,$ we have three identities:

$\begin{align} a &= b\cdot\cos\gamma + c\cdot\cos\beta\\ b &= c\cdot\cos\alpha + a\cdot\cos\gamma \\ c &= a\cdot\cos\beta + b\cdot\cos\alpha . \end{align}$

The identities are obtained by drawing the altitudes - one at a time, and applying the definition of the cosine in the two right triangles so obtained. For each side, Euclid would consider two cases, as he did in II.12 and II.13. With the extension of the cosine functions from its original definition for acute angles, we may combine the acute and obtuse cases in one identity.

Let's multiply the first identity by $a,$ the second by $b,$ and the third by $c,$ and subtract the first two from the third:

$c^{2} - a^{2} - b^{2} = -2ab\cdot\cos\gamma ,$

which is exactly the Law of Cosines.

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