Law of Cosines

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Law of Cosines

The law of Cosines is an equation that we can use to solve for missing angles or sides of a triangle.


Form 1:

a 2 = b 2 + c 2 2 b c cos A

Form 2:

b 2 = a 2 + c 2 2 a c cos B

Form 3:

c 2 = a 2 + b 2 2 a b cos C

Example 1: Find length of segment A B ¯

step 1:
Determine which form of the Law of Cosines to use based on given information

Given Information
Angles Sides
A = unknown a = 6.7
B = unknown b = 5.0
C = 65° c = answer

Because the question is asking us to find A B ¯ or side c, we will use form 3 of the equation.

c 2 = a 2 + b 2 2 a b cos C

step 2:
rewrite equation in terms of c instead of c by taking the square root of both sides of the equation.2

c 2 = a 2 + b 2 2 a b cos C
c 2 = a 2 + b 2 2 a b cos ( C )
c = a 2 + b 2 2 a b cos ( C )

step 3:
Plug in known values to the equation found in step 2

c = 6.7 2 + 5.0 2 2 (6.7) (5.0) cos ( 65° )

Final Answer:

c = 6.45

Example2: Find the measure of angle B

step 1:
Determine which form of the Law of Cosines to use based on given information

Given Information
Angles Sides
A = unknown a = 3.7
B = answer b = 6.5
C = unknown c = 8.0
Because the question is asking us to find angle B, we will use form 2 of the equation.

b 2 = a 2 + c 2 2 a c cos B

step 2:
Rewrite the equation to isolate the answer, B.

b 2 = a 2 + c 2 2 a c cos B b 2 a 2 c 2 = 2 a c cos B b 2 a 2 c 2 2 a c = cos B

To isolate B and get rid of Cos( ) we apply the inverse trig function which is arcos or cos-1 in this case.

cos 1 ( b 2 a 2 c 2 2 a c ) = cos 1 ( cos B ) cos 1 ( b 2 a 2 c 2 2 a c ) = B

step 3:
Plug in known values to equation found in step 2

cos 1 ( 6.5 2 3.7 2 8.0 2 2 (3.7) (8.0) ) = B

Final Answer:

B = 52.23 °