Derivation of Cosine Law

Tags: 
derivation of formula
trigonometry
cosine law

The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.
 

$ a^2 = b^2 + c^2 - 2bc\cos A $

$ b^2 = a^2 + c^2 - 2ac\cos B $

$ c^2 = a^2 + b^2 - 2ab\cos C $

 

Derivation:
Figure for the derivation of Cosine LawConsider the triangle to the right:

Cosine function for triangle ADB
$ \cos A = \dfrac{x}{c} $

$ x = c\cos A $
 

Pythagorean theorem for triangle ADB
$ x^2 + h^2 = c^2 $

$ h^2 = c^2 - x^2 $
 

Pythagorean theorem for triangle CDB
$ (b - x)^2 + h^2 = a^2 $
 

Substitute h2 = c2 - x2
$ (b - x)^2 + (c^2 - x^2) = a^2 $

$ (b^2 - 2bx + x^2) + (c^2 - x^2) = a^2 $

$ b^2 - 2bx + c^2 = a^2 $
 

Substitute x = c cos A
$ b^2 - 2b(c \cos A) + c^2 = a^2 $
 

Rearrange:

$ a^2 = b^2 + c^2 - 2bc\cos A $

 

The other two formulas can be derived in the same manner.
 

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