Cosine rule

In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states





c

2


=

a

2


+

b

2


−
2
a
b
cos
⁡
γ
,


{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma ,}
where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c. For the same figure, the other two relations are analogous:





a

2


=

b

2


+

c

2


−
2
b
c
cos
⁡
α
,


{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha ,}






b

2


=

a

2


+

c

2


−
2
a
c
cos
⁡
β
.


{\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta .}
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90 degrees, or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem:





c

2


=

a

2


+

b

2


.


{\displaystyle c^{2}=a^{2}+b^{2}.}
The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.

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