The Cosine Law
A. The Cosine Law
In Lesson 4.4 you saw a relationship between two sides and two angles of a triangle.
In this lesson you will explore how three sides and one angle in a triangle are related.
Open the Cosine Law applet (12 February 2013, Created with GeoGebra) to begin exploring this relationship.
In the applet you are given two expressions for the triangle:
Notice that all three sides, a, b, and c, are used and one angle, A, is used.
Move the points around to make various triangles. How are the two expressions, mentioned above, related for any triangle you make?
While using the Cosine Law applet you may have noticed that the two expressions were always equal. From using the applet, it seems reasonable that the two relationships can be written as 
. This is, in fact, true and will be proven in Example 1. This relationship is often called the cosine law.
Example 1
Prove the Cosine Law.
Begin with triangle ABC.
Draw a perpendicular to b through B. Label the perpendicular h and the two parts of b as p and q.
There are four relationships that will be used later in the proof. You will explain why each of these relationships is true in Lesson 4.5 Game On!
The following completes the proof of the cosine law. You will be asked to explain each step in Lesson 4.5 Game On!
