An elementary proof of the cosine law

High School Geometry, High School Mathematics, High School Trigonometry, Problem Solving and Proofs

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.

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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 Theorem, a ^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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