Ever since I first learned it, I’ve always loved the cos rule. It says that if a triangle has two sides a and b, with an angle of C between them, then the remaining side c can be found in this way: c^{2} = a^{2} + b^{2} – 2ab cos C.

The reason I like it is that it’s like Pythagoras’ Theorem, but with an adjustment to make it work out right for non-right-angles. See the c^{2} = a^{2} + b^{2} is Pythagoras’ theorem and the – 2ab cos C is the bit that adjusts the answer to take into account that the angle C is more or less than a right angle.

Anyway, I recently saw Euclid’s original version of the cos rule — and it doesn’t use cos at all!

It goes something like this (this is my version of course — I’m sure Euclid didn’t use this exact turn of phrase):

You have a triangle ABC with sides a, b and c opposide their matching angles.

Find the perpendicular from point B to the line b, and label D the point where it meets b. Label d the distance from C to D, and say that d is negative if it’s outside the triangle and positive if it’s inside the triangle.

Then c^{2} = a^{2} + b^{2} – 2bd.

Here’s a diagram of the two situations:

Isn’t that cool? The coolness of it is that it’s all based on the lengths you can construct from the triangle, and there’s no need to make reference to a strange function of angles like cos in order to do it. In fact, all you need to *prove* it is pythagoras’ theorem, which you can convince yourself is true just using paper and scissors.

Somehow it makes the cos rule — and even cos itself — even more real to know it has such a grounding in the lengths and areas of triangles.