Where does the cosine rule come from?

For the mathematical puritans out there the cosine rule derives from the following:

(you really don’t need to know this)

Sine and Cosine are used to find the angle of a shape

Find “a” KNOWING the angle A and length b and c.

Redraw the diagram

Redraw to understand the cosine rule

Pythagoras tells us that in triangle I , $x^2+d^2=b^2$ ................ equation 1

Trigonometry tells us that $cosA=x/b$

Therefore $x=bcosA$ ………… equation 2

Pythagoras also tells us that $(c-x)^2+d^2=a^2$ …………… equation 3

Multiplying equation 3 out we get: $(c-x)(c-x)+d^2=a^2$

$c^2-cx-cx+x^2+d^2=a^2$

$c^2-2cx+x^2+d^2=a^2$ ……………. equation 4

Substitute equation 1 into equation 4 we get:

$c^2-2cx+b^2=a^2$

Replacing $x$ with equation 2 we get:

$c^2-2c(bcosA)+b^2=a^2$

Rearrange slightly

$a^2=b^2+c^2-2cb\cosA$

But again we emphasise that you really don’t need this.