Law of Cosines (LC)

The law of cosines is more difficult to use and normally are used to solve part of a problem
and then utilize the law of sines.

If angle 1 = 100 ⁰ side A = 46 cm and side C = 35 cm
What is the length of angle 3, angle 2 and side B

B² = A² + C² - 2AC cos ∠1

B² = 46² + 35² - 2(46)(35) cos 100⁰

B² = 2116 + 1225 - 3220(-.173648)

B² = 3900.147132

√B² = √3900.147132

B = 62.451 cm

Now if we wish we can use the Law of Sines to find the other angles.
Since we are still learning the Law of Cosines lets use them.

C² = A² + B² - 2AB cos ∠2

35² = 46² + 62.451² - (2)46(62.451) cos ∠2

1225 = 2116 + 3900.127 - 5745.492 cos ∠2

-4791.127 = -5745.492 cos ∠2

.833893251 = cos ∠2

33.499⁰ = ∠2



A² = B² + C² - 2BC cos ∠3

46² = 62.451² + 35² - (2)(62.451)(35) cos ∠3

2116 = 39000.127 + 1225 - 4371.57cos ∠3

-3009.127 = - 4371.57cos ∠3

.688340116 = cos ∠3

46.5⁰ = ∠3

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