Derivation of the Cosine Law

Introduction

In the previous post, we have discussed an elementary proof of the sine law. In this post, we derive the cosine law. Just like the sine law, the cosine law relates the sides and angles of a triangle.

cosine law

The cosine law states that for any triangle ABC, 

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 proof is as follows.

Theorem

Let ABC be a triangle,

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.

Proof

Let h be the height of triangle ABC as shown in the figure above. Triangle BCD is a right triangle, so by the Pythagorean theorem,

a^2 = (c - x)^2 + h^2

a^2 = c^2 -2cx + x^2 + h^2 (1).

However, in triangle ADC

b^2 = x^2 + h^2

therefore, substituting in (1), we have

a^2 = c^2 - 2cx + b^2 (2).

Also, in triangle ADC,

\cos A = \frac{x}{b}

so, x = b \cos A.  Substituting in (2), we have

a^2 = c^2 - 2c b \cos A + b^2.

Rearranging the terms on the equation, we have,

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

The proof above can be also used to derive the other two equations.

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