Cosine

Imagine a triangle which has sides of length a, b and c.

The angles within the triangle are α, β and γ (where α is opposite a, and so on).

Now, drop a perpendicular from α onto side a, such that it splits a into lengths x and y (with x on the side closer to angle γ).

Then, we can see that:

	cosβ = y / c	and	cosγ = x / b
∴	y = c cosβ	and	x = b cosγ
∴	a = b cosγ + c cosβ
∴	a2 = ab cosγ + ac cosβ     [1]

Now, in our triangle, we could also drop a perpendicular from β to b and from γ to c, and by going through the same process as above, we would end up with similar equations, but with:

a→b→c→a and α→β→γ→α in one, and

a→c→b→a and α→γ→β→α in the other.

Resulting in:

b² = bc cosα + ab cosγ     [2]     and

c² = ac cosβ + bc cosα     [3]

Now, if we take the three equations, and do [3] - [2] - [1], we have:

c² - b² - a² = ac cosβ + bc cosα - bc cosα - ab cosγ - ab cosγ - ac cosβ

∴ c² = a² + b² - 2ab cosγ

And this is the cosine rule. I have to confess that this is a small, but clearly important, piece of maths that I had either completely forgotten, or never come across in the first place.

The reason for mentioning it here is that at the weekend I spent a while working on a question from Jake's maths homework for him, and finally managed to come up with the answer.

Jake was pleased with my help, but came back to me a few minutes later and pointed out that it could have been done in a fraction of the time that I had taken, by the simple application of the cosine rule!

Oh dear. I sense that I am going to start struggling to keep up very soon ....