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In ABC, let a, b and c be the lengths of sides opposite to the angles A, B and C respectively.
Draw a perpendicular from vertex A onto side BC. And call it AM.
Let BM = x, so that MC = a-x.
In ABM and ACM we can write using Pythagorus's theorem,
AM2 = c2-x2
AM2 = b2-(a-x)2
c2 - x2 = b2 - (a - x)2
c2 - x2 = b2 - a2 +2ax -x2
c2 = b2 - a2 +2ax
x = (c2 - b2 + a2)/2a
cosB = x/c
= (c2 - b2 + a2)/2ac
= (a2 + c2 - b2 )/2ac
Similarly we can write the expression for cosA and cosC.
It's called cosine rule.
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