Where does the $-2ab$ term come from in the cosine law?

Question

I understand that in the cosine rule i.e. $c^2 = a^2 + b^2 - 2ab \\cos C$, the cosine function acts to bring down the value of $c^2$ for acute angles ($\\cos C>0$, $-2ab\\cos C<0$ ) and increase the value of $c^2$ for obtuse angles ($\\cos C <0$, $-2ab\\cos C > 0$). I still wonder where the $2ab$ term comes from? Any ideas about the intuition behind that? Regards,

Answer 1

When you draw a triangle with vertices A, B, and C, and with the opposite sides labeled a, b, and c, draw an altitude from A to the opposite side a. Call that altitude h. The altitude creates two right triangles. For the leg of the triangle with b as its hypotenuse, we can call that side x. The leg of the other right triangle will then be a-x.

Using the Pythagorean Theorem, the two right triangles have the following relationships:

(a-x)²+h²=c²

h²+x²=b²

Multiply out the first equation to get a^2-2ax+x²+h²=c². Next substitute b² in for h²+x² to get the equation a²-2ax+b²=c².

Now for the triangle with the leg we labeled x, we know cos C=x/b. Solving that equation for x gives us x=b(cos C). We can then substitute that into the last equation to yield a²-2ab(cos C)+b²=c², or the Law of Cosines:

c²=a²+b²-2ab(cos C)