Proof of Cosine Rule
Below is the proof by Pythagoras theorem of the cosine rule, a2=b2+c2- 2bccosA.
This assumes you understand Pythagoras theorem (visit Pythagoras theorem to view my lesson on it), how to use basic trigonometry(basic trigonometry lesson). If you want to learn how to use the cosine and sine rule, opposed to just learning the proof) visit by sine and cosine rule page.
The proof is done using the letters of the following triangle
and we are trying to prove the cosine rule:
a2=b2+c2- 2bccosA
In triangle CBL
a2 = (c-x)2 + h2
a2 = c2 – 2cx + x2 + h2
h2 = a2 -c 2- x2 + 2cx <<EQN1
in triangle CLA
b2 = h2 + x2
h2 = b2 – x2 <<EQN2
eqn1 – eqn2 :: 0 = a2 – c2 – b2 +2cx
a2 = c 2+ b2 – 2cx <<EQN3
in CLA
cosA = x/b
x = bcosA
in eqn3
a2 = c2 + b2 – 2bccosA
So there is the proof for the cosine rule using Pythagoras theorem.
Comments Left
i need a chinese proof of pythagoras rule
WordPress generated a link here from my blog:
http://newtonexcelbach.wordpress.com/2008/05/18/elegant-proofs-1/
This is a very simple proof of Pythagoras’ theorem. It’s not in Chinese, but it has very few words, so would be easy to translate
I’m trying to justify your “magic number” of 17.817. I guess I don’t understand where you’re coming from, as the 12th root of 2 is not the above figure, but closer to 1.05946309 – - -. What am I misunderstanding?
Unfortunately the diagram of the triangle isn’t showing, and it’s hard to follow the description without it.