# What is the law of cosines?

Dec 21, 2014

Cosider the triangle:

(Picture source: Wikipedia)

you can relate the sides of this triangle in a kind of "extended" form of Pitagora's Theorem giving:

${a}^{2} = {b}^{2} + {c}^{2} - 2 b c \cdot \cos \left(\alpha\right)$
${b}^{2} = {a}^{2} + {c}^{2} - 2 a c \cdot \cos \left(\beta\right)$
${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cdot \cos \left(\gamma\right)$

As you can see you use this law when your triangle is not a right-angled one.

Example:
Consider the above triangle in which:
$a = 8 c m$
$c = 10 c m$
beta=60° therefore:
${b}^{2} = {a}^{2} + {c}^{2} - 2 a c \cdot \cos \left(\beta\right)$
b^2=8^2+10^2-2*8*10*cos(60°) but cos(60°)=1/2
so: ${b}^{2} = 84 \mathmr{and} b = \sqrt{84} = 9 , 2 c m$