An elementary proof of the cosine law

The cosine law states that in triangle ABC with side lengths a, b, and c, the following equations are satisfied:

a ^2 = b^2 + c^2 - 2bc \cos A*

b ^2 = a^2 + c^2 - 2ac \cos B

c ^2 = a^2 + b^2 - 2ab \cos C

The discussion below shows how these equations were derived.


Consider triangle ABC with side lengths a, b, and c. Drop an altitude h from C and let D be the intersection of h and AB as shown in the figure below.  If we let x be the length of AD, then BD = c - x.

(1) Remembering the mnemonic SOHCAHTOA, in a triangle, the cosine of an angle is equal to length of the side adjacent to it divided by the length of the hypotenuse. Therefore, in triangle ACD,  \cos A = \displaystyle\frac{x}{b}. Simplifying, we have x = b \cos A.

(2) In triangle BDC, by the Pythagorean Theorema ^2 = h^2 + (c - x)^2 which when simplified equals to a ^2 = h^2 + c^2 - 2cx + x^2.

(3) Also, in triangle ACD, by the Pythagorean theorem, b^2 = x^2 + h^2.

(4) Rearranging the equation in step (2), we have a ^2 = c^2 - 2cx + (x^2 + h^2).  Now, substituting the equation in step (3) to the preceding equation and rearranging the terms, we have a ^2 = b^2 + c^2 - 2cx .

(5) Substituting the equation in step (1) to the equation in step (4), we have a ^2 = b^2 + c^2 - 2bc \cos A, which is equal to the first equation above *.

The formula for b^2 and c^2 follow from the proof above. This proves the cosine law.

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