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Law of Cosines


c^2=a^2+b^2-2ac*Cos(C)

a^2=b^2+c^2-2ac*Cos(A)

b^2= a^2+c^2-2ac*Cos(B)

To solve for cosines:

cos C= (a^2+b^2-c^2)/(2*a*b)

cos A= (b^2+c^2-a^2)/(2*b*c)

cos B= (a^2+c^2-b^2)/(2*a*c)

Examples

In triangle ABC:

If angle C is 120�, side b is 50 cm, and side a is 70cm, find the length of hypotenuse c.

c^2=a^2+b^2-2*a*b*cos(C)

c^2=70^2+50^2-2*70*50*cos(120)

c^2=10900cm

c=10900^(1/2)*

c=104.4 cm

* ^(1/2) is the square root. It is important that you put the 1/2 in parentheses.


In triangle ABC:

If side a is equal to 10 cm, side b is equal to 5 cm, and side c is equal to 12 cm, find the measure of angle C.

cosC= a^2+b^2-c^2/(2*a*b)

cosC=(10^2+5^2-12^2)/(2*5*10)

cosC=-19/100

cosC= -.19

cos^-1C=101�

When angle C=90�, the equation is reduced to c^2=a^2+b^2

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