Malorie's Blog

9-4 Law of Cosines

in 9-4, we are learned how to solve triangles using law of cosines.
you use this when you do not have a right angle or you can't use any of the other formulas. Law of Cosines is more like a last resort.

The formula is
(opposite leg) squared= (adjacent leg) squared + (adjacent leg) squared - 2(adjacent leg)(adjacent leg)Cos(angle between)

Example:

Two legs of a triangle are 2 and 6. The angle between the is 100 degrees. Find the other side.

you are looking for the opposite of 100 degrees, you can't use law of sines because you do not have any opposite lengths.

x^2=2^2 + 6^2 - 2(2)(6)Cos 100

To solve this you put the WHOLE equation under the square root thing or your answer wont come out right.

x=6.65


Example 2:

The lengths of a triangle are 3, 4, and 5. Find all angles.

First, we'll find angle A.
(you can find whatever you choose first)

4^2= 5^2 + 3^2 - 2(5)(3)Cos A

4^2 - 5^2 - 3^2= -2(5)(3)Cos A

divide by cos A, then take the inverse of cos to get and angle measure.

A=53.1

now, you can either do the same for the rest of the measures or use law of sines.
I prefer to use law of sines because i now have an opposite.

Sin B/5 = Sin 53.1/4

Cross multiply and then divide 4, then take the inverse of Sin B

B = 88.4

Now that you have two angles you can just subtract them from 180 to get your third angle

180-88.4-53.1 = 38.5

C= 38.5