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Law of Cosines

One side of a triangle when the opposite angle and two sides are known.

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The "Law of Cosines" can be used to calculate one side of a triangle when the angle opposite and the other two sides are known.

Law of cosines

The "Law of Cosines" can be expressed as

$$ c^2 = a^2 + b^2 - 2 a  b cos C            \tag{1} $$

where

a, b and c = length of triangle sides (m, ft ..)

C = angle opposite side c (degrees)

Example - Calculate Side in Triangle

If side a = 1 m, side b = 1.3 m and angle C = 60 degrees - the side c can be calculated by modifying eq. 1 to 

$$ c = \sqrt{ (1 m)^2 + (1.3 m)^2 - 2 (1 m) (1.3 m) cos (60 degrees) }  = \underline{1.18} m         $$

Law of Cosines Calculator

The generic calculator below can used to calculate the side c:

Law of Cosines - Excel Template

Law of cosines - excel template

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