Derivation of 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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