1. The Law of Cosines

2. Let's consider types of triangles with the three pieces of information shown below.
We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles.
SAS
AAA
You may have a side, an angle, and then another side
You may have all three angles.
This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down"
SSS
You may have all three sides

3. LAW OF COSINES Law of Cosines
Use these to find missing sides and angles
LAW OF COSINES C B A C a b B A c

4. Solve a triangle where b = 1, c = 3 and A = 80° B
Example 1.
Solve a triangle where b = 1, c = 3 and A = 80° B 3
Draw a picture. a
80 ° A C
This is SAS 1 A
minus 2 times the product of those other sides
times the cosine of the angle between those sides
One side squared
sum of each of the other sides squared
Do we know an angle and side opposite it?
No, so we must use Law of Cosines.
Hint: we will be solving for the side opposite the angle we know.

5. a = 2.99 Example 1.Solve a triangle where b = 1, c = 3 and A = 80° B a
80 ° A 1 A
minus 2 times the product of those other sides
times the cosine of the angle between those sides
sum of each of the other sides squared
One side squared
a = 2.99

6. B = 18.85° C = 81.15° We'll label side a with the value we found. B 
We now have all of the sides but how can we find an angle?  3
18.85°
2.99
80°
Hint: We have an angle and a side opposite it.
81.15°  A C 1 C
Angle  is easy to find since the sum of the angles is a triangle is 180° C
= 18.85
B = 18.85°
C = 81.15°

7. Solve a triangle where a = 5, b = 8 and c = 9
Example #2
Solve a triangle where a = 5, b = 8 and c = 9
This is SSS
Draw a picture. B
Do we know an angle and side opposite it?
No, so we must use Law of Cosines. 9 5
84.3 C A 8 C
Let's use largest side to find largest angle first.

8. C = 84.3° Example #2 B 9 5 84.3  A C 8 C C C C C One side squared
minus 2 times the product of those other sides
times the cosine of the angle between those sides
sum of each of the other sides squared
One side squared C C C C

9. A = 33.5° B = 62.2° Example #2 B    A C B A B B9
62.2
Do we know an angle and side opposite it? 5  
33.5
84.3 A C
Yes, so use Law of Sines. 8 B A
A = 33.5° B B
B = 62.2°