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:
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
cosA = x/b
x = bcosA
a2 = c2 + b2 – 2bccosA
So there is the proof for the cosine rule using Pythagoras theorem.