The Law of Cosines

Written by tutor Jeffery D.

In this article we will prove the Law of Cosines and explain how it can be used to determine side lengths and angles of triangles in two particular cases.

The Law of Cosines can be used to “solve” triangles in two particular situations. Recall that to “solve” triangles means to determine unknown side lengths or angle measures. The Law of Cosines is used in the following two cases:

    1. Two side lengths and the included angle are known.
    2. Three side lengths are known.

Theorem: Law of Cosines – The square of one side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of their included angle.

In symbols, this theorem is:

c2 = a2 + b2 – 2ab cos C

Figure 1

Proof. We begin by constructing a triangle (see Fig. 1) on the xy-plane such that the vertex of angle C is at the origin and side b lies along the positive x-axis.The vertex of angle A has coordinates (b, 0) and the coordinates of the vertex of the angle B has coordinates (a cos C, a sin C). The reason that the angle B has such coordinates can be seen by recalling how sine and cosine are
defined on the unit circle. We can determine the length of side c by using the distance formula to compute the distance between the vertex of angles B and A. Recall the distance formula is given by

Plugging in the coordinates of our vertices

*In the above equation, we can see that cos2θ + sin2θ = 1.This is the Pythagorean identity for trigonometric functions.

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